LinearAlgebra_160423_01

© Art Traynor 2011
Mathematics
Definition
Mathematics
Wiki: “ Mathematics ”
1564 – 1642
Galileo Galilei
Grand Duchy of Tuscany
( Duchy of Florence )
City of Pisa
Mathematics – A Language
“ The universe cannot be read until we have learned the language and
become familiar with the characters in which it is written. It is written
in mathematical language…without which means it is humanly
impossible to comprehend a single word.
Without these, one is wandering about in a dark labyrinth. ”

© Art Traynor 2011
Mathematics
Definition
Algebra – A Mathematical Grammar
Mathematics
A formalized system ( a language ) for the transmission of
information encoded by number
Algebra
A system of construction by which
mathematical expressions are well-formed
Expression
Symbol Operation Relation
Designate expression
elements or Operands
( Terms / Monomials )
Transformations or LOC’s
capable of rendering an
expression into a relation
A mathematical Structure
between operands
represented by a well-formed
Expression
A well-formed symbolic representation of Operands ( Terms or Monomials ) ,
of discrete arity, upon which one or more Operations ( Laws of Composition - LOC’s )
may structure a Relation
1. Identifies the explanans
by non-tautological
correspondences
Definition
2. Isolates the explanans
as a proper subset from
its constituent
correspondences
3. Terminology
a. Maximal parsimony
b. Maximal syntactic
generality
4. Examples
a. Trivial
b. Superficial
Mathematics
Wiki: “ Polynomial ”
Wiki: “ Degree of a Polynomial ”

© Art Traynor 2011
Mathematics
Disciplines
Algebra
One of the disciplines within the field of Mathematics
Mathematics
Others are Arithmetic, Geometry,
Number Theory, & Analysis

The study of expressions of symbols ( sets ) and the
well-formed rules by which they might be manipulated
to preserve validity .

Algebra
Elementary Algebra
Abstract Algebra
A class of Structure defined by the object Set and
its Operations ( or Laws of Composition – LOC’s )

Linear Algebra
Mathematics

© Art Traynor 2011
Mathematics
Definitions
Expression
Transformations or LOC’s
capable of rendering an
expression into a relation
A mathematical structure
between operands represented
by a well-formed expression
A well-formed symbolic representation of Operands ( Terms or Monomials ) ,
of discrete arity, upon which one or more Operations ( LOC’s ) may structure a Relation
Expression – A Mathematical Sentence
Proposition
A declarative expression
asserting a fact, the truth
value of which can be
ascertained
Formula
A concise symbolic
expression positing a relation
VariablesConstants
An alphabetic character
representing a number the
value of which is arbitrary,
unspecified, or unknown
Operands ( Terms / Monomials )
A transformation
invariant scalar quantity
Mathematics
Predicate
A Proposition admitting the
substitution of variables
O’Leary, Section 2.1,
Pg. 41
Expression constituents consisting of Constants and
Variables exhibiting exclusive parity
Polynomial
An Expression composed of Constants ( Coefficients ) and Variables ( Unknowns) with
an LOC’s of Addition, Subtraction, Multiplication and Non-Negative Exponentiation

© Art Traynor 2011
Mathematics
Definitions
Expression
Transformations capable of
rendering an expression
into a relation
A mathematical structure between operands represented
by a well-formed expression
Proposition
the truth value of which can
be ascertained
Formula
A concise symbolic
expression positing a relation
VariablesConstants
An alphabetic character
representing a number the
value of which is arbitrary,
unspecified, or unknown
Operands ( Terms / Monomials )
A transformation
invariant scalar quantity
Equation
A formula stating an
equivalency class relation
Inequality
A formula stating a relation
among operand cardinalities
Function
A Relation between a Set of inputs and a Set of permissible
outputs whereby each input is assigned to exactly one output
Univariate: an equation containing
only one variable
( e.g. Unary )
Multivariate: an equation containing
more than one variable
( e.g. n-ary )
Mathematics
Expression constituents consisting of Constants and
Variables exhibiting exclusive parity
Polynomial

© Art Traynor 2011
Mathematics
Definitions
Expression
Proposition Formula
VariablesConstants
Operands ( Terms )
Equation
A formula stating an
equivalency class relation
Linear Equation
An equation in which each term is either
a constant or the product of a constant
and (a) variable[s] of the first degree
Mathematics
Polynomial

© Art Traynor 2011
Mathematics
Expression
Mathematical Expression
A representational precursive discrete composition to a
Mathematical Statement or Proposition ( e.g. Equation )
consisting of :

Operands / Terms
Expression
A well-formed symbolic
representation of Operands
( Terms or Monomials ) ,
of discrete arity, upon which one
or more Operations ( LOC’s ) may
structure a Relation
Mathematics
n Scalar Constants ( i.e. Coefficients )
n Variables or Unknowns
The Cardinality of which is referred to as the Arity of the Expression
Constituent representational Symbols composed of :
Algebra
Laws of Composition ( LOC’s )
Governs the partition of the Expression
into well-formed Operands or Terms
( the Cardinality of which is a multiple of Monomials )

© Art Traynor 2011
Mathematics
Arity
Arity
Expression
The enumeration of discrete symbolic elements ( Variables )
comprising a Mathematical Expression
is defined as its Arity

The Arity of an Expression can be represented by
a non-negative integer index variable ( ℤ + or ℕ ),
conventionally “ n ”

A Constant ( Airty n = 0 , index ℕ )or Nullary
represents a term that accepts no Argument

A Unary expresses an Airty n = 1
A relation can not be defined for
Expressions of Arity less than
two: n < 2
A Binary expresses Airty n = 2
All expressions possessing Airty n > 1 are n-ary, Multary, Multiary, or Polyadic
VariablesConstants
Operands
Expression
Polynomial

© Art Traynor 2011
Mathematics
Expression
Arity
Operand
 Arithmetic : a + b = c
The distinct elements of an Expression
by which the structuring Laws of Composition ( LOC’s )
partition the Expression into discrete Monomial Terms
 “ a ” and “ b ” are Operands
 The number of Variables of an Expression is known as its Arity
n Nullary = no Variables ( a Scalar Constant )
n Unary = one Variable
n Binary = two Variables
n Ternary = three Variables…etc.
VariablesConstants
Operands
Expression
Polynomial
n “ c ” represents a Solution ( i.e. the Sum of the Expression )
Arity is canonically
delineated by a Latin
Distributive Number,
ending in the suffix “ –ary ”

© Art Traynor 2011
Mathematics
Arity
Arity ( Cardinality of Expression Variables )
Expression
A relation can not be defined for
Expressions of Arity less than
two: n < 2
Nullary
Unary
n = 0
n = 1
Binary n = 2
Ternary n = 3
1-ary
2-ary
3-ary
Quaternary n = 4 4-ary
Quinary n = 5 5-ary
Senary n = 6 6-ary
Septenary n = 7 7-ary
Octary n = 8 8-ary
Nonary n = 9 9-ary
n-ary
VariablesConstants
Operands
Expression
Polynomial
0-ary

© Art Traynor 2011
Mathematics
Operand
Parity – Property of Operands
Parity
n is even if $ k | n = 2k
n is odd if $ k | n = 2k+1
Even  Even
Integer Parity
Same Parity
Even  Odd Opposite Parity

© Art Traynor 2011
Mathematics
Polynomial
Expression
A well-formed symbolic
representation of operands, of
discrete arity, upon which one
or more operations can
structure a Relation
Expression
Polynomial Expression
A Mathematical Expression ,
the Terms ( Operands ) of which are a compound composition of :
Polynomial
Constants – referred to as Coefficients
Variables – also referred to as Unknowns
And structured by the Polynomial Structure Criteria ( PSC )
arithmetic Laws of Composition ( LOC’s ) including :
Addition / Subtraction
Multiplication / Non-Negative Exponentiation
LOC ( Pn ) = { + , – , x bn ∀ n ≥ 0 }
An excluded equation by
Polynomial Structure Criteria ( PSC )
Σ an xi
n
i = 0
P( x ) = an xn + an – 1 xn – 1 +…+ ak+1 xk+1 + ak xk +…+ a1 x1 + a0 x0
Variable
Coefficient
Polynomial Term
From the Greek Poly meaning many,
and the Latin Nomen for name





© Art Traynor 2011
Mathematics
Degree
Expression
Polynomial
Degree of a Polynomial
Polynomial
The Degree of a Polynomial Expression ( PE ) is supplied by that
of its Terms ( Operands ) featuring the greatest Exponentiation
For a multivariate term PE , the Degree of the PE is supplied by that
Term featuring the greatest summation of Variable exponents

P = Variable Cardinality & Variable Product
Exponent Summation
& Term Cardinality
Arity
Latin “ Distributive ” Number
suffix of “ – ary ”
Degree
Latin “ Ordinal ” Number
suffix of “ – ic ”
Latin “ Distributive ” Number
suffix of “ – nomial ”
0 =
1 =
2 =
3 =
Nullary
Unary
Binary
Tenary
Constant
Linear
Quadratic
Cubic
Monomial
Binomial
Trinomial
An Expression composed of
Constants ( Coefficients ) and
Variables ( Unknowns) with an
LOC of Addition, Subtraction,
Multiplication and Non-
Negative Exponentiation

© Art Traynor 2011
Mathematics
Degree
Polynomial
Nullary
Unary
p = 0
p = 1 Linear
Binaryp = 2 Quadratic
Ternaryp = 3 Cubic
1-ary
2-ary
3-ary
Quaternaryp = 4 Quartic4-ary
Quinaryp = 5 5-ary
Senaryp = 6 6-ary
Septenaryp = 7 7-ary
Octaryp = 8 8-ary
Nonaryp = 9 9-ary
“ n ”-ary
Arity Degree
Monomial
Binomial
Trinomial
Quadranomial
Terms
Constant
Quintic
P
Septic
Octic
Nonic
Decic
Sextic
aka: Heptic
aka: Hexic

© Art Traynor 2011
Mathematics
Degree
Expression
Polynomial
Polynomial
An Expression composed of
Constants ( Coefficients ) and
Variables ( Unknowns) with an
LOC of Addition, Subtraction,
Multiplication and Non-
Negative Exponentiation
For a PE with multivariate term(s) ,
the Degree of the PE is supplied by
that Term featuring the greatest summation
of individual Variable exponents

P( x ) = ai xi
0 Nullary Constant Monomial
P( x ) = ai xi
1
Unary Linear Monomial
P( x ) = ai xi
2
Unary Quadratic Monomial
ai xi
1 yi
1P( x , y ) =
Binary Quadratic Monomial
Univariate
Bivariate

© Art Traynor 2011
Mathematics
Degree
Expression
Polynomial
Polynomial
For a multivariate term PE , the Degree of the PE is supplied by that
Term featuring the greatest summation of Variable exponents

P( x ) = ai xi
0 Nullary Constant Monomial
P( x ) = ai xi
1
Unary Linear Monomial
P( x ) = ai xi
2
ai xi
1 yi
1P( x , y ) = Binary Quadratic Monomial
ai xi
1 yi
1zi
1P( x , y , z ) = Ternary Cubic Monomial
Univariate
Bivariate
Trivariate
Multivariate

© Art Traynor 2011
Mathematics
Quadratic
Expression
Polynomial
Quadratic Polynomial
Polynomial
A Unary or greater Polynomial
composed of at least one Term and :
Degree precisely equal to two
Quadratic ai xi
n ∀ n = 2
 ai xi
n yj
m ∀ n , m n + m = 2|:
Etymology
From the Latin “ quadrātum ” or “ square ” referring
specifically to the four sides of the geometric figure
Wiki: “ Quadratic Function ”
Arity ≥ 1
 ai xi
n ± ai + 1 xi + 1
n ∀ n = 2
Binary Quadratic Monomial
Unary Quadratic Binomial
 ai xi
n yj
m ± ai + 1 xi + 1
n ∀ n + m = 2 Binary Quadratic Binomial

© Art Traynor 2011
Mathematics
Equation
Equation
Expression
An Equation is a statement or Proposition
( aka Formula ) purporting to express
an equivalency relation between two Expressions :

Expression
Proposition
asserting a fact whose truth
value can be ascertained
Equation
A symbolic formula, in the form of a
proposition, expressing an equality relationship
Formula
A concise symbolic
expression positing a
relationship between
quantities
VariablesConstants
Operands
Symbols
Operations
The Equation is composed of
Operand terms and one or more
discrete Transformations ( Operations )
which can render the statement true
( i.e. a Solution )
Polynomial

© Art Traynor 2011
Mathematics
Equation
Solution
Solution and Solution Sets
 Free Variable: A symbol within an expression specifying where
a substitution may be made
Contrasted with a Bound Variable
which can only assume a specific
value or range of values
 Solution: A value when substituted for a free variable which
renders an equation true
Analogous to independent &
dependent variables
Unique Solution: only one solution
can render the equation true
(quantified by $! )
General Solution: constants are
undetermined
value-specified (bound?)
Unique Solution
Particular Solution
General Solution
Solution Set
n A family (set) of all solutions –
can be represented by a parameter (i.e. parametric representation)
 Equivalent Equations: Two (or more) systems of equations sharing
the same solution set
Section 1.1, (Pg. 3)
Any of which could include a Trivial Solution

© Art Traynor 2011
Mathematics
Equation
Solution
Solution and Solution Sets
Unique Solution: only one solution
can render the equation true
(quantified by $! )
undetermined
value-specified (bound?)
Solution Set
n For some function f with parameter c such that
f(xi , xi+1 ,…xn – 1 , xn ) = c
the family (set) of all solutions is defined to include
all members of the inverse image set such that
f(x) = c  f -1(c) = x
f -1(c) = {(ai , ai+1 ,…an-1 , an )  Ti· Ti+1 ·…· Tn-1· Tn | f(ai , ai+1 ,…an-1 , an ) = c }
where Ti· Ti+1 ·…· Tn-1· Tn is the domain of the function f
o f -1(c) = { }, or empty set ( no solution exists )
o f -1(c) = 1, exactly one solution exists ( Unique Solution, Singleton)
o f -1(c) = { cn } , a finite set of solutions exist
o f -1(c) = {∞ } , an infinite set of solutions exists
Inconsistent
Consistent
Section 1.1,
(Pg. 5)

© Art Traynor 2011
Mathematics
Linear Equation
Linear Equation
Equation
An Equation consisting of:
Operands that are either
Any Variables are restricted to the First Order n = 1
Linear Equation
An equation in which each term
is either a constant or the
product of a constant and (a)
variable[s] of the first order
Expression
Proposition
Equation
Formula
n Constant(s) or
n A product of Constant(s) and
one or more Variable(s)
The Linear character of the Equation derives from the
geometry of its graph which is a line in the R2 plane

As a Relation the Arity of a Linear Equation must be
at least two, or n ≥ 2 , or a Binomial or greater Polynomial

Polynomial

© Art Traynor 2011
Mathematics
Equation
Linear Equation
Linear Equation
 An equation in which each term is either a constant or the product
of a constant and (a) variable[s] of the first order
Term ai represents a Coefficient
b = Σi= 1
n
ai xi = ai xi + ai+1 xi+1…+ an – 1 xn – 1 + an xn
Equation of a Line in n-variables
 A linear equation in “ n ” variables, xi + xi+1 …+ xn-1 + xn
has the form:
n Coefficients are distributed over a defined field
( e.g. N , Z , Q , R , C )
Term xi represents a Variable ( e.g. x, y, z )
n Term a1 is defined as the Leading Coefficient
n Term x1 is defined as the Leading Variable
Coefficient = a multiplicative factor
(scalar) of fixed value (constant)

© Art Traynor 2011
Mathematics
Linear Equation
Equation
Standard Form ( Polynomial )
 Ax + By = C
 Ax1 + By1 = C
For the equation to describe a line ( no curvature )
the variable indices must equal one

 ai xi + ai+1 xi+1 …+ an – 1 xn –1 + an xn = b
 ai xi
1 + ai+1 x 1 …+ an – 1 x 1 + a1 x 1 = bi+1 n – 1 n n
ℝ
2
: a1 x + a2 y = b
ℝ
3
: a1 x + a2 y + a3 z = b
Blitzer, Section 3.2, (Pg. 226)
Test for Linearity
 A Linear Equation can be expressed in Standard Form
As a species of Polynomial , a Linear Equation
can be expressed in Standard Form
 Every Variable term must be of precise order n = 1
Linear Equation
An equation in which each term
is either a constant or the
product of a constant and (a)
variable[s] of the first order
Expression
Proposition
Equation
Formula
Polynomial

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Linear Equation – Solution Consistency
Unique Solution
Particular Solution
General Solution
Solution Set
n A family (set) of all solutions – can be represented by a parameter
No Solution - Inconsistent
1
0
0
2
1
0
– 1
0
0
4
3
– 2
Represents “ 0 = – 2 ” ,
a contradiction,
and thus no solution {  }
to the LE system for which
the augmented matrix stands
1x1 + 0x2 – 3x3 = – 1
System
0x1 + 1x2 – 1x3 = 0
x1 – 3x3 = – 1
System
x2 = x3

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Linear Equation – Solution Consistency
Solution Set - Consistent
n A family (set) of all solutions – can be represented by a parameter
1x1 + 0x2 – 3x3 = – 1
System
0x1 + 1x2 – 1x3 = 0
x1 = 3x3 – 1
System
x2 = x3
Note that x3 can be parameterized
( as a composite function f ○ g → ( f ○ g )( x ) = f ( g ( x )) with y = f ( u ) and u = g ( x3 ) )
x2 = x3 = u Tautology/Identity*
x1 = 3u – 1 The solution set for “ f(u) ” can thus be indexed by/over Z+
representing a countably infinte solution set
*

© Art Traynor 2011
Mathematics
‘ f(x) ’
‘– f ’
‘ x ’
‘ f(x) ’
‘ +f -1 ’
‘ x ’
Linear Algebra
Solution
Linear Equation – Solution Set
Solution Set
n For some function f with parameter c such that
f ( xi , xi+1 ,…xn – 1 , xn ) = c
the family ( set ) of all solutions is defined to include
all members of the inverse image set such that
f ( x ) = c  f -1( c ) = x

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
aij xi + aij+1 xi+1 + . . . + ain – 1 xn – 1 + ain xn = bi
ai+1j xi + ai+1j+1 xi+1 + . . . + ai+1n – 1 xn – 1 + ai+1n xn = bi+1
am – 1j xi + am – 1j+1 xi+1 + . . . + am – 1n – 1 xn – 1 + am – 1n xn = bm – 1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
amj xi + amj+1 xi+1 + . . . + amn xn – 1 + amn xn = bm
Linear Equation – System
 A system of m linear equations in n variables
is a set of m equations ,
each of which is linear in the same n variables
Linear Equation System
Solution Set
The set S = { si , si+1 ,…sn-1 , sn } which renders
each of the equations in the system true

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Linear Equation – Back Substitution
x – 2y = 5
y = – 2
System
x – 2 y = 5
x – 2 (– 2 ) = 5
x + 4 = 5
x = 5 – 4
x = 1
Solution Set – Singleton, Unique Solution, ( exactly one solution )
S = { 1, – 2 }

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Linear Equation Equivalence
 Equivalent Linear Equations: Two (or more) systems of
linear equations sharing the same solution set
 Gaussian Elimination
Operations Producing Linear Equation Equivalent Systems
Permutation/Interchange – of two equations
Multiply – an equation by a non-zero constant
Add – a multiple of an equation to another equation
Otherwise known as Elementary Row Operations Section 1.2, (Pg. 14)
n ERO’s should always proceed with an Augend/Multiplicand of lesser
rank and Summand/Multiplier of greater rank ( Aij < Amn ) yielding
a Sum/Product substituted for the second Operand
n Multiplication by a scalar ( non-zero constant ) need not affect any change
in rank for the resultant row

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Linear Equation – Row Echelon Form (REF)
1
0
0
2
1
0
– 1
0
1
4
3
– 2
 A matrix in Row-Echelon Form ( REF ) has three distinguishing characteristics
Any rows consisting entirely of zeros is positioned at the bottom of the matrix
For each row that does not consist entirely of zeros, the first non-zero entry is
a “ 1 ” ( called the Leading One, aka Pivot )

1
0
0
2
1
0
– 1
0
1
4
3
– 2
0 0 0 0
0 0 0 0
Section 1.2, (Pg. 16) Every matrix is row equivalent to a matrix in Row-Echelon Form ( REF )

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Linear Equation – Row Echelon Form (REF)
 A matrix in Row-Echelon Form ( REF ) has three distinguishing characteristics
For two successive (non-zero) rows, the leading one in the higher row is farther
to the left than the leading one ( Pivot ) in the lower row

1
0
0
2
1
0
– 1
0
1
4
3
– 2
0 0 0 0
 A matrix in Reduced Row-Echelon Form ( RREF ) has one additional characteristic
Every column that has a leading one ( Pivot ) has zeros in every position
above and below its leading one ( Pivot )

1
0
0
0
1
0
0
0
1
4
3
– 2
0 0 0 0
 Every matrix is row equivalent to a matrix in Row-Echelon Form ( REF )

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Linear Equation – Reduced Row Echelon Form (RREF)
1
– 1
2
– 2
3
– 5
3
0
5
9
– 4
17
x – 2y + 3z = 9
– x + 3y = – 4
2x – 5y + 5z = 17
System Augmented Matrix
R2 :
R2 + R1  R2´
– 1 3 0 – 4
+ R1 : 1 – 2 3 9
= R2´ : 0 1 3 5
1
0
2
– 2
1
– 5
3
3
5
9
5
17
R3 – 2R1  R3´

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
1
2
– 2
– 5
3
5
9
17
Augmented Matrix
R3 :
– 2R1 : – 2 4 – 6 – 18
= R3´ : 0 – 1 – 1 – 1
1
0
0
– 2
1
– 1
3
3
– 1
9
5
– 1
R3 + R2  R3´´
0 1 3 5 R3 – 2R1  R3´
2 – 5 5 17

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Augmented Matrix
R3 :
+ R2 :
= R3´´ : 0 0 2 4
1
0
– 2
1
3
3
9
5
R3 + R2  R3´´
1 – 2 3 9
0 1 3 5
0 – 1 – 1 – 1
0 – 1 – 1 – 1
0 1 3 5
0 0 2 4
R3  R3´´ ´1
2

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Augmented Matrix
R3 : 1 – 2 3 9
0 1 3 5
R3  R3´´ ´1
2
1 – 2 3 9
0 1 3 5
0 0 2 4
R3 :
1
2
R3´´ ´ :
0 0 2 4
0 0 1 2
0 0 1 2 0 0 1 2
Matrix is in
Row-Echelon Form ( REF )

Proceeding to
Reduced
Row-Echelon Form
( RREF )
R1 + 2R2  R1´

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Augmented Matrix
R1 :
+ 2R2 :
R1´ :
0 2 6 10
1 0 9 19 1 0 9 19
0 1 3 5
0 0 1 2
1 – 2 3 9
0 1 3 5
0 0 1 2
R1 + 2R2  R1´
1 – 2 3 9
R2 – 3R3  R2´´

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Augmented Matrix
R2 :
– 3R3 :
R2´´ :
0 0 – 3 – 6
0 1 0 – 1
1 0 9 19
0 1 0 – 1
0 0 1 2
R1 + 9R3  R1´´
1 0 9 19
0 1 3 5
0 0 1 2
R2 – 3R3  R2´´
0 1 3 5

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Augmented Matrix
R1 :
– 9R3 :
= R1´´ :
0 0 – 9 – 18
1 0 0 1
0 1 0 – 1
0 0 1 2
R1 – 9R3  R1´´
1 0 9 19
0 1 0 – 1
0 0 1 2
1 0 0 1
1 0 9 19
Matrix is in
Reduced
Row-Echelon Form
( RREF )

© Art Traynor 2011
Mathematics
Matrices
Matrix
For positive integers m and n, an m x n (“m by n”) matrix is a rectangular array
populated by entries aij , located at the i-th row and the j-th column:
Linear Algebra
 M = N: the matrix is a square of order n
 The a11 , a22 , a33 , amn , sequence of entries is the
main diagonal ( ↘ ) of the matrix
M = # of Rows
i = Row Number Index
N = # of Columns
j = Column Number Index
Row 1 a11
Row 2
Row 3
Row m
.
.
.
a21
a31
am1
.
.
.
a12
a22
a32
am2
.
.
.
a13
a23
a33
am3
.
.
.
. . .
. . .
. . .
. . .
.
.
.
a1n
a2n
a3n
amn
.
.
.
mi
mi+1
mi+2
mm
nj nj+1 nj+2 nn
C1 C2 C3 . . . C4

© Art Traynor 2011
Mathematics
Matrices
Matrix
For positive integers m and n, an m x n (“m by n”) matrix is a rectangular array
Row 1
populated by entries aij , located at the i-th row and the j-th column:
Linear Algebra
a11
Row 2
Row 3
Row m
.
.
.
a21
a31
am1
.
.
.
a12
a22
a32
am2
.
.
.
a13
a23
a33
am3
.
.
.
. . .
. . .
. . .
. . .
.
.
.
a1n
a2n
a3n
amn
.
.
.
mi
mi+1
mi+2
mm
nj nj+1 nj+2 nn
M = # of Rows
N = # of Columns
C1 C2 C3 . . . Cn
 “ i ” is the row subscript
 “ j ” is the column subscript

© Art Traynor 2011
Mathematics
Linear Algebra
Definitions
Diagonal Matrix
A matrix Anx n is said to be diagonal when all of the entries outside the
main diagonal ( ↘ ) are zero
The matrix Dnx n is diagonal if :
dij = 0 if i ≠ j "ij  { dii , di+1,i+1 ,…, dn–1,n–1 , dnn }


Any square diagonal matrix is also a Symmetric Matrix
A diagonal matrix is also both Upper-Triangular and Lower-Triangular
The Identity Matrix In is a diagonal matrix
Any square Zero Matrix is a diagonal matrix
d11
d21
d31
dm1
.
.
.
d12
d22
d32
dm2
.
.
.
d13
d23
d33
dm3
.
.
.
. . .
. . .
. . .
. . .
.
.
.
d1n
d2n
d3n
dmn
.
.
.
Dnx n =

© Art Traynor 2011
Mathematics
Linear Algebra
Definitions
Tr ( A ) = Σi = 1
n
aii = a11 + a22 +…+ an –1n – 1 + ann
Matrix Trace
The trace of a matrix Anx n is the sum of the main diagonal entries Section 2.1, (Pg. 50)
a11
a21
a31
am1
.
.
.
a12
a22
a32
am2
.
.
.
a13
a23
a33
am3
.
.
.
. . .
. . .
. . .
. . .
.
.
.
a1n
a2n
a3n
amn
.
.
.
A

© Art Traynor 2011
Mathematics
Matrices
Linear Algebra
1
– 1
2
– 4
3
0
3
– 1
– 4
5
– 3
6
x – 4y + 3z = 5
– x + 3y – z = – 3
2x – 4z = – 6
System Augmented Matrix
1
– 1
2
– 4
3
0
3
– 1
– 4
Coefficient Matrix
M = # of Rows
N = # of Columns
Augmented Matrix
 A matrix representing a system of linear equations including both
the coefficient and constant terms
Coefficient = a multiplicative factor
(scalar) of fixed value (constant)
Coefficient Matrix
 A augmented matrix excluding any constant terms and populated
only by the variable coefficients

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Gaussian Elimination With Back-Substitution
 Express the system of linear equations as an Augmented Matrix
Interchange – of two equations
Add – a multiple of an equation to another equation
Every matrix is row equivalent to a matrix in Row-Echelon Form ( REF ) Section 1.2, (Pg. 16)
 Apply ERO’s to restate the matrix in Row Echelon Form (REF)
 Use Back Substitution to solve for unknown variables Section 1.1, (Pg. 6)
Order Matters! Operate from left-to-right

© Art Traynor 2011
Mathematics
Linear Algebra
Solution
Gauss-Jordan Elimination
 Follow steps 1 & 2 of Gaussian Elimination
Every matrix is row equivalent to a matrix in Row-Echelon Form ( REF ) Section 1.2, (Pg. 16)
Apply ERO’s to restate the matrix in Row Echelon Form (REF)
 Use Back Substitution to solve for unknown variables Section 1.1, (Pg. 6)
Order Matters! Operate from left-to-right

Express the system of linear equations as an Augmented Matrix
n Interchange – of two equations
n Multiply – an equation by a non-zero constant
n Add – a multiple of an equation to another equation
 Keep Going!
Continue to apply ERO’s until matrix assumes
Reduced Row Echelon Form ( RREF )
 Section 1.2, (Pg. 15)

© Art Traynor 2011
Mathematics
Linear Algebra
Homogeneity
Homogenous Systems of Linear Equations
A linear equation system in which each of the constant terms is zero
aij xi + aij+1 xi+1 + . . . + ain – 1 xn – 1 + ain xn = 0
ai+1j xi + ai+1j+1 xi+1 + . . . + ai+1n – 1 xn – 1 + ai+1n xn = 0
am – 1j xi + am – 1j+1 xi+1 + . . . + am – 1n – 1 xn – 1 + am – 1n xn = 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
amj xi + amj+1 xi+1 + . . . + amn xn – 1 + amn xn = 0
 A homogenous LE system Must have At Least One Solution Section 1.2, (Pg. 21)
Every homogenous LE system is Consistent
# Equations < # Variables  Infinitely Many Solutions

© Art Traynor 2011
Mathematics
Matrix Representation
Linear Algebra
Matrix Representation Methods
 Uppercase Letter Designation
A , B , C
 Bracket-Enclosed Representative Element
[ aij ] , [ bij ] , [ cij ]
a11
a21
a31
am1
.
.
.
a12
a22
a32
am2
.
.
.
a13
a23
a33
am3
.
.
.
. . .
. . .
. . .
. . .
.
.
.
a1n
a2n
a3n
amn
.
.
.
 Rectangular Array
Brackets denote a Matrix ( i.e. not a specific element/real number)

© Art Traynor 2011
Mathematics
Matrix Equality
Linear Algebra
Matrix Equality Section 1.2, (Pg. 40)
A = [ aij ]
B = [ bij ]
are equal
when
Amxn = Bmxn
aij = bij
1 ≤ i ≤m
1 ≤ j ≤n
a1 a2 a3 . . . ana =
C1 C2 C3 . . . Cn
Row Matrix / Row Vector
A 1 x n (“ 1 by n ”) matrix is a single row
Column Matrix / Column Vector
b1
b2
b3
bm
.
.
.
C1
An m x 1 (“ m by 1 ”) matrix is a single column

© Art Traynor 2011
Mathematics
Matrix Operations
Linear Algebra
Matrix Summation Section 1.2, (Pg. 41)
A = [ aij ]
B = [ bij ]
is given
by
+ A + B = [ aij + bij ]
– 1
0
2
1
1
– 1
3
2
+ = ( – 1 + 1 ) =
( 0 + [ – 1] )
( 2 + 3 )
( 1 + 2 )
0
– 1
5
3
Scalar Multiplication
1
– 3
2
2
0
1
4
– 1
2
A = 3A =
3 ( 1 ) 3 ( 2 ) 3 ( 4 )
3 ( – 3 ) 3 ( 0 ) 3 ( – 1 )
3 ( 2 ) 3 ( 1 ) 3 ( 2 )
3
– 9
6
6
0
3
12
– 3
6
3A =

© Art Traynor 2011
Mathematics
Matrix Operations
Linear Algebra
Section 1.2, (Pg. 42)Matrix Multiplication
– 1
4
5
3
– 2
0
A =
A = [ aij ] Amx n
B = [ bij ] Bnx p
then AB = [ cij ] = Σk = 1
n
aik bkj = ai 1 b1 j + ai2 b2j +…+ ain –1 bn-1j + ain bnj
The entries of Row “ Aik” ( the i-th row ) are multiplied by the entries of “ Bkj” ( the j-th column )
and sequentially summed through Row “ Ain” and Column “ Bnj” to form the entry at [ cij ]
– 3
– 4
2
1
B =
c11 c12
C = c21 c22
c31 c32
a11b11 + a12b21 a11b12 + a12b22
= a21b11 + a22b21 a21b12 + a22b22
a31b11 + a32b21 a31b12 + a32b22
Product Summation Operand Count
For Each Element of AB (single entry)
Product Summation (Column-Row) Index
 For the product of two matrices to be defined, the column count of the multiplicand matrix
must equal the row count of the multiplier matrix ( i.e. Ac = Br )

ABmx p

© Art Traynor 2011
Mathematics
Systems Of Linear Equations
Linear Algebra
Linear Equation System
a11 x1 + a12 x2 + a13 x3 = b1
a21 x1 + a22 x2 + a23 x3 = b2
a31 x1 + a32 x2 + a33 x3 = b3
Matrix-Vector Notation
a11 a13
Ax = b  a21 a23
a31 a33
=
a12
a22
a32
A
x1
x2
x3
x
b1
b2
b3
b

© Art Traynor 2011
Mathematics
Systems Of Linear Equations
Linear Algebra
Partitioned Matrix Form (PMF)
Ax = b  =
A x
b
a11
a21
am1
.
.
.
a12
a22
am2
.
.
.
. . .
. . .
. . .
.
.
.
a1n
a2n
amn
.
.
.
x1
x2
xn
.
.
.
Ax = b  =
ai1
b
a11
a21
am1
.
.
.
x1
ai2
a12
a22
am2
.
.
.
+ x2 + . . . + xn
ain
a1n
a2n
amn
.
.
.
Ax = b  =
Ax
b
a11 x1
a21 x1
am1 x1
.
.
.
+ a12 x2 +
+ a22 x2 +
+ am2 x2 +
.
.
.
. . .
. . .
. . .
.
.
.
+ a1n xn
+ a2n xn
+ amn xn
.
.
.
Ax = x1 a1 + x2 a2 + . . . + xn an = b

© Art Traynor 2011
Mathematics
Section 2.1 Review
Linear Algebra
Section 2.1 Review
 Introduce Three Basic Martrix Operations
Matrix Addition
Scalar Multiplication
Matrix Multiplication

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Properties Of Matrices – Addition & Scalar Multiplication
Commutative
(Addition)
Associative
(Addition)A +( B + C ) = ( A +B ) + C
Changes Order of Operations
as per “PEM-DAS”, Parentheses
are the principal or first operation
A +B = B +A
Re-Orders Terms
Does Not Change
Order of Operations – PEM-DAS
Associative
(Multiplication)( cd ) A = c ( dA )
Distributive
( Scalar Over Matrix Addition )c ( A + B ) = c A + cB
Distributive
( Scalar Addition Over
Matrix Addition )
( c + d ) A = c A + dA

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Proofs
Let A = [ aij ], B = [ bij ]
 Introduce/Define/Declare The Constituents to be Proven
This statement declares A & B to be Matrices
Specifies the row & column count index variables

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Properties Of Matrices – Identities & Zero Matrices
Multiplicative Identity
Multiplicative Zero Identity
1A = A
Additive IdentityA + 0mx n = A
Additive InverseA + ( – A ) = 0mx n
c A = 0mx n
if c = 0
or A = 0mx n

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Properties Of Matrices – Matrix Multiplication
Distributive
( LHS )
A( BC ) = ( AB ) C
Order of Terms is preserved
Affects Order of Operations
Sequence – PEM-DAS
Distributive
( RHS )
Associative
( Scalar Over Matrix
Multiplication )
A( B + C ) = AB + AC
( A + B ) C = AC + BC
c ( AB ) = ( c A )B
= A ( c B )
Associative
(Multiplication)
AC = BC
CA = CB
( C is invertible )
Right Cancellation Property
A = B
if then
Left Cancellation Property

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Properties Of Matrices – Proofs
A( BC ) = ( AB ) C
Associative
(Multiplication)
Σk = 1
n
T = [ tij ] = ( aik bkj )ckjΣk = 1
n
Σi = 1
n
( yj )Σj = 1
n
( xi )
Σi = 1
n
( xi ) ( y1 + y2 +…+ yn –1 + yn )
( x1 + x2 +…+ xn –1 + xn ) y1 + ( x1 + x2 +…+ xn –1 + xn ) y2 +…
( x1 + x2 +…+ xn –1 + xn ) yn –1 + ( x1 + x2 +…+ xn –1 + xn ) yn
x1 y1 + x2 y1 +…+ xn –1 y1 + xn y1 + x1 y2 + x2 y2 +…+ xn –1 y2 + xn y2 +…
x1 yn –1 + x2 yn –1 +…+ xn –1 yn –1 + xn yn –1 + x1 yn + x2 yn +…+ xn –1 yn + xn yn

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
A( BC ) = ( AB ) C
Associative
(Multiplication)
Σk = 1
n
T = [ tij ] = ( aik bkj )ckjΣk = 1
n
Σi = 1
n
( yj )Σj = 1
n
( xi )
Σi = 1
n
( xi ) ( y1 + y2 +…+ yn –1 + yn )
( x1 + x2 +…+ xn –1 + xn ) y1 + ( x1 + x2 +…+ xn –1 + xn ) y2 +…
( x1 + x2 +…+ xn –1 + xn ) yn –1 + ( x1 + x2 +…+ xn –1 + xn ) yn
x1 y1 + x2 y1 +…+ xn –1 y1 + xn y1 + x1 y2 + x2 y2 +…+ xn –1 y2 + xn y2 +…
x1 yn –1 + x2 yn –1 +…+ xn –1 yn –1 + xn yn –1 + x1 yn + x2 yn +…+ xn –1 yn + xn yn
Σi = 1
n
Σj = 1
n
xi yj

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
A = [ aij ] Amx n
B = [ bij ] Bnx p
then AB = [ cij ] = Σk = 1
n
aik bkj = ai 1 b1 j + ai2 b2j +…+ ain –1 bn-1j + ain bnj
Product Summation Operand Count
Product Summation (Column-Row) Index
ABmx p
The entries of Row “ Aik” ( the i-th row ) are multiplied by the entries of “ Bkj” ( the j-th column )
and sequentially summed through Row “ Ain” and Column “ Bnj” to form the entry at [ cij ]

ai,1 b1, j + ai,1 b1, j +1 ai,1 b1,n – 1 + ai,1 b1,n
ai+1,2 b2, j + ai+1,2 b2, j +1 ai+1,2 b2, n – 1 + ai+1,2 b2, n
.
.
.
.
.
.
.
.
.
+ . . .+
+ . . .+
.
.
.
.
.
.
an – 1,n – 1bn – 1, j + an – 1,n – 1bn – 1, j +1 an – 1,n – 1bn – 1, n – 1 + an – 1,n – 1bn
an,nbn, j + an,n bn, j +1 an,n bn,n – 1 + an,n bn,n
+ . . .+
+ . . .+
For Each Element of AB (single entry)

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Identity Matrix
For Amx n

A In = A
A Im = A
Matrix Exponentiation
For Ak = AA…A
K factors

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Transpose Matrix
a11
a21
am1
.
.
.
a12
a22
am2
.
.
.
. . .
. . .
. . .
.
.
.
a1n
a2n
amn
.
.
.
A =
a11
a12
a1n
.
.
.
a21
a22
a2n
.
.
.
. . .
. . .
. . .
.
.
.
am1
am2
amn
.
.
.
AT =
1
2
0
2
1
0
0
0
1
C =
1
2
0
2
1
0
0
0
1
CT =
Symmetric Matrix: C = CT
If C = [ cij ] is a symmetric matrix, cij = cji for i ≠ j
C = [ cij ] is a symmetric matrix, Cmx n = CT
nx p for m = n = p

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
1
2
0
2
1
0
0
0
1
C =
1
2
0
2
1
0
0
0
1
CT =
If C = [ cij ] is a symmetric matrix, cij = cji , "i,j | i ≠ j
C = [ cij ] is a symmetric matrix, Cmx n = CT
nx p , "m,n, p | m = n = p
Symmetric Matrix
A Symmetric Matrix is a
Square Matrix that is
equal to it Transpose ( e.g. Cmx n = CT
mx n , "m,n | m = n)


© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Properties Of Matrices – Transposes
( AT ) T = A
Transpose of a
Scalar Multiple
( A + B ) T = A T + B T
( c A ) T = c ( A T )
Transpose of a Transpose
Transpose of Sum
( AB ) T = B T A T Transpose of a Product
Reverse Order of Terms
( interchange multiplicand & multiplier
terms in the product expression )
Symmetry of A
Matrix & The Product
of Its Transpose
AAT = ( AAT ) T
ATA is also symmetric

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Matrix Inverse
A matrix Anx n is Invertible or Non-Singular when
$ matrix Bnx n | AB = BA = In
In : Identity Matrix of Order n
Bnx n : The Multiplicative Inverse of A
A matrix that does not have an inverse is Non-Invertible or Singular
Non-square matrices do not have inverses
n For matrix products Amx n Bnx p where m ≠ n ≠ p,
AB ≠ BA as [ aij ≠ bij ] ??



© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Matrix Inverse
There are two methods for determining an inverse matrix A-1
of A ( if an inverse exists):
Solve Ax = In for X

Adjoin the Identity Matrix In (on RHS ) to A forming the doubly-
augmented matrix [ A In ] and perform EROs concluding in RREF to
produce an [ In A-1 ] solution

A test for determining whether an inverse matrix A-1 of A exists:
Demonstrate that either/or AB = In = BA
Section 2.3 (Pg. 64)

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Matrix Inverse
Uniqueness Property
If A is invertible, then its inverse is Unique
Notation: The inverse of A is denoted as A-1
If A is invertible, then the LE system represented by Ax = b
has a Unique Solution given by x = A– 1 b


© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Matrix Inverse – by Matrix Equation
x11 x12
x21 x22
1
– 1
4
– 3
+ =
A x
1
0
0
1
In
For a coefficient matrix Anx n the A-1
nx n matrix is that whose product
yields a solution matrix to the corresponding In identity matrix

1x11 +
– 1x21 +
4x21
( – 3x21 )
=
Ax
1x11 +
– 1x21 +
4x21
( – 3x21 )
1
0
0
1
In
1x11 + 4x21 = 1
– 1x21 + ( – 3x21 ) = 0
1x11 + 4x21 = 0
– 1x21 + ( – 3x21 ) = 1  – 3
1
– 4
1
A-1Ax = In Ax = In

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Matrix Inverse – by Gauss-Jordan Elimination
x11 x12
x21 x22
1
– 1
4
– 3
+ =
A x
1
0
0
1
In
An invertible coefficient Anx n matrix can be combined with its corresponding
xnx n unknown/variable matrix to form an Axnx n = In equation matrix

This equation matrix is composed itself of identical coefficient
column vectors

1 x11 +
– 1 x21 +
4 x21
( – 3 x21 )
=
Ax
1 x11 +
– 1 x21 +
4 x21
( – 3 x21 )
1
0
0
1
In

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra

column vectors

1 x11 +
– 1 x21 +
4 x21
( – 3 x21 )
=
Ax
1 x11 +
– 1 x21 +
4 x21
( – 3 x21 )
1
0
0
1
In
1x11 + 4x21 = 1
– 1x21 + ( – 3x21 ) = 0
1x11 + 4x21 = 0
– 1x21 + ( – 3x21 ) = 1
Ax = In Ax = In
Rather than solve the two column equation vectors separately,
they can be solved simultaneously by adjoining the identity
matrix to the shared coefficient matrix


© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Matrix Inverse – by Gauss-Jordan Elimination ( GJE )

column vectors

1 x11 + 4 x21 = 1
– 1 x21 + ( – 3 x21 ) = 0
1 x11 + 4 x21 = 0
– 1 x21 + ( – 3 x21 ) = 1
Ax = In Ax = In
Rather than solve the two column equation vectors separately,
they can be solved simultaneously by adjoining the identity
matrix to the shared coefficient matrix…

1
– 1
4
– 3
A
1
0
0
1
In
…then execute ERO’s to effect a GJ-Elimination of the
“ doubly augmented ” [ A I ] matrix the conclusion of
which will yield an [ I A-1 ] inverse matrix

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra

column vectors

1 x11 + 4 x21 = 1
– 1 x21 + ( – 3 x21 ) = 0
1 x11 + 4 x21 = 0
– 1 x21 + ( – 3 x21 ) = 1
Ax = In Ax = In
The adjoined, “ doubly-augmented ” coefficient matrix , by means of ERO’s , is reduced by
GJ-Elimination to produce the [ I A-1 ] inverse matrix

1
– 1
4
– 3
A
1
0
0
1
In
 – 3
1
– 4
1
A-1
1
0
0
1
In
Which is confirmed by verifying either of the following
n AA-1 = I
n AA-1 = A-1 A

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Matrix Inverse – 2x2 Matrix ( Special Case )
For a square matrix A2x 2 , given by:
The inverse A-1 of the root matrix A2x 2 is given by the following product:
a
c
b
d = ad – cb
d
– c
– b
a
The difference of the diagonal products forms the
multiplicand denominator of the matrix whose
product yields the inverse of the root matrix
1
ad – cb
A-1 = NegateSwitcheroo
Abstract Algebra,
Lecture 2 @ 18:30
The scalar multiple is the
inverse of the root matrix
Determinant!
The “multiplier” matrix is a
half-negated permutation of
the root matrix!

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Properties Of Inverse Matrices
A : An Invertible Matrix
k : A positive integer , Z+
c : A non-zero scalar, c ≠ 0
A– 1
Ak
c A
AT
are Invertible
and the following are true:
( A– 1 ) – 1 = A
( Ak ) – 1 = A– 1 A– 1 … A– 1 = ( A– 1 ) k
K factors
( cA ) – 1 = A– 1 1
c
( AT ) – 1 = ( A– 1 ) T
Aj Ak = Aj+k
( Aj ) k = Ajk
( AB ) – 1 = B– 1 A– 1 ( B is also invertible )

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Properties of Matrix Exponentiation
Aj Ak = Aj+k
( Aj ) k = Ajk

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Elementary Matrices
An Elementary Matrix, Anx n is:
A square matrix ( n x n )
Obtained from a corresponding Identity Matrix In

Results from a single Elementary Row Operation ( ERO )
If E is an Elementary Matrix, then:
E is obtained from an ERO on a corresponding Identity Matrix Im

EA is the product of the same ERO performed on an Am x n matrix
Matrices Amx n & Bmx n are Row Equivalent when:
$ a finite set of Elementary Matrices E1 , E2 ,… , Ek such that
B = Ek Ek – 1 … E2 E1 A

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Elementary Matrices - Properties
If E is an Elementary Matrix then:
E– 1 exists
E– 1 is an Elementary Matrix
A square matrix A is invertible if-and-only-if:
A can be expressed as the product of elementary matrices
Every Elementary Matrix has an inverse
Matrix Equivalency conditions, for Anx n matrix:
“ A ” is invertible
Ax = b has a unique solution for every n x 1 column matrix b
Ax = 0 has only the trivial solution
“ A ” is row-equivalent to In

“ A ” can be written as the product of elementary matrices

© Art Traynor 2011
Mathematics
Linear Algebra
Definitions
Upper & Lower Triangular Matrices
a11
a21
a31
am1
.
.
.
0
a22
a32
am2
.
.
.
0
0
a33
am3
.
.
.
. . .
. . .
. . .
. . .
.
.
.
0
0
0
amn
.
.
.
L
For an Anx n square matrix:
“ L ” is a lower triangular matrix where all entries above the Main Diagonal
are zero, and only the lower half is populated with non-zero entries.

a11
0
0
0
.
.
.
a12
a22
0
0
.
.
.
a13
a23
a33
0
.
.
.
. . .
. . .
. . .
. . .
.
.
.
a1n
a2n
a3n
amn
.
.
.
U
“ U ” is a lower triangular matrix where all
entries below the Main Diagonal are zero, and
only the upper half is populated with non-zero
entries.


© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
LU Factorization
A square matrix Anx n can be written as a product A = LU if:
“ L ” is a lower triangular matrix where all entries above the Main Diagonal are
zero, and only the lower half is populated with non-zero entries, and …

“ U ” is a lower triangular matrix where all entries below the Main Diagonal are
zero, and only the upper half is populated with non-zero entries


© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Determinants
Every square matrix Anx n can be associated with a real number
defined as its Determinant

Notation: det ( A ) = |a |
a11 x1 + a12 x2 = b1
a21 x1 + a22 x2 = b2
Example:
2-LE System with (2) unknowns  yields solutions with common denominators
b1 a22 – b2 a12
a11 a22 – a21 a12
x1 =
b2 a11 – b1 a21
a11 a22 – a21 a12
x2 =
Determinant of a 2 x 2 Matrix
a11 a12
a21 a22
A = = det ( A ) = |A | = a11 a22 – a21 a12
a11
a21
a12
a22
= a11a22 – a21a12
Determinant is the difference
of the product of the diagonals
The Determinant is a
polynomial of Order “ n ”

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Determinants
Every square matrix Anx n can be associated with a real number
defined as its Determinant

Notation: det ( A ) = |a |
a11 x1 + a12 x2 = b1
a21 x1 + a22 x2 = b2
Example:
b1 a22 – b2 a12
a11 a22 – a21 a12
x1 =
b2 a11 – b1 a21
a11 a22 – a21 a12
x2 =
Determinant of a 2 x 2 Matrix
Determinant is the difference
of the product of the diagonals
The Determinant is a
polynomial of Order “ n ”
a
c
b
d = ad – cb
The Determinant is the Area
(an n-Manifold ) of the
parallelogram suggested by
the addition of the vectors
represented by the matrix

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Minors & Cofactors
For a square matrix Anx n

The Minor Mij
of the entry aij
is the determinant of the matrix
obtained by deleting the ith row and jth column of A

The Cofactor Cij
of the entry aij
is Cij = ( – 1 )i+j Mij

Example:
a11 a13
a21 a23
a31 a33
a12
a22
a32
Minor of a21
a11 a13
a21 a23
a31 a33
a12
a22
a32
Minor of a22
a12
a32
a13
a33
, M21 =
a11
a31
a13
a33
, M22 =
A Minor IS A DETERMINANT!!

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Minors & Cofactors

The Cofactor Cij
of the entry aij

Example:
a11 a13
a21 a23
a31 a33
a12
a22
a32
Minor of a21
a11 a13
a21 a23
a31 a33
a12
a22
a32
Minor of a22
a12
a32
a13
a33
, M21 =
a11
a31
a13
a33
, M22 =
Cofactor of a21 Cofactor of a22
C21 = ( – 1 )2+1 M21 = – M21 C22 = ( – 1 )2+2 M22 = M22

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Determinant of a Square Matrix
For a square matrix Anx n of order n ≥ 2 , then:
The Determinant of A is
the sum of the entries in the first row of A
multiplied by their respective Cofactors

det ( A ) = |A | = a1j C1j = a11 C11 + a12 C12 +…+ a1, n –1 Cn, n –1 + a1n C1nΣj = 1
n
The process of determining this sum is Expanding The Cofactors ( in the first row )

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Expansion By Cofactors
For a square matrix Anx n of order n , the determinant of A is given by:
An ith row expansion
det ( A ) = |A | = aij Cij = ai1 Ci1 + ai2 Ci2 +…+ ai, n –1 Ci, n –1 + ain CinΣj = 1
n
A jth column expansion
det ( A ) = |A | = aij Cij = a1j C1j + a2j C2j +…+ an –1, j Cn –1, j + anj CnjΣj = 1
n

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Determinants – 3x3 Matrix ( Special Case )
For a square matrix A3x 3 , the determinant of A is given by:
The first two columns are adjoined to the RHS of the matrix
a11 a13
a21 a23
a31 a33
a12
a22
a32
a11
a21
a31
a12
a22
a32
Product sums are formed by first multiplying along the main diagonal proceeding to the right
a11 a13
a21 a23
a31 a33
a12
a22
a32
a11
a21
a31
a12
a22
a32
➀ ➁ ➂
= a11a22 a33 + a12a23 a31 + a13a21 a32 = UD
Upper
Diagonal

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Determinants – 3x3 Matrix ( Special Case )
For a square matrix A3x 3 , the determinant of A is given by:
Remaining product differences are then formed by multiplying along the LHS bottom diagonal
proceeding to the right

a11 a13
a21 a23
a31 a33
a12
a22
a32
a11
a21
a31
a12
a22
a32
➃ ➄ ➅
= UD – a31a22 a13 – a32a23 a11 – a33a21 a12
= UD – LD Upper
Diagonal
minus
Lower
Diagonal

© Art Traynor 2011
Mathematics
Linear Algebra
Definitions
Diagonal Matrix
A matrix Anx n is said to be diagonal when all of the entries outside the
main diagonal ( ↘ ) are zero
The matrix Dnx n is diagonal if :
dij = 0 if i ≠ j "ij  { dii , di+1,i+1 ,…, dn–1,n–1 , dnn }


d11
d21
d31
dm1
.
.
.
d12
d22
d32
dm2
.
.
.
d13
d23
d33
dm3
.
.
.
. . .
. . .
. . .
. . .
.
.
.
d1n
d2n
d3n
dmn
.
.
.
Dnx n =
A matrix Anx n that is both upper AND lower triangular is said to be
diagonal

The determinant of a triangular matrix Dnx n is the product of its main diagonal elements
det ( D ) = |D | = aii = a11 a22 … a n –1, n –1 ain

Πi = 1
n

© Art Traynor 2011
Mathematics
Linear Algebra
Definitions
EROs & Determinants (Properties)
Permutation:
[ A ]  Pij  [ B ]
det ( B ) = – det ( A )
|B | = – | A |

Multiplication by a Scalar:
[ A ]  cRi  [ B ]
det ( B ) = c det ( A )
|B | = c | A |

Addition to a Row Multiplied by a Scalar:
[ A ]  Ri + cRj  [ B ]
det ( B ) = det ( A )
|B | = | A |

There are three “effects” to a resultant
matrix which are unique to each of the
three EROs
Permutation
Row Addition

© Art Traynor 2011
Mathematics
Linear Algebra
Definitions
Zero Determinants
A matrix Anx n will feature a determinant of zero
det ( A ) = 0
|A | = 0
if any of the following pertain

One row/column of “ A ” consists of all zeros
Two rows/columns of “ A ” are equal
One row/column of “ A ” is a multiple of another

© Art Traynor 2011
Mathematics
Linear Algebra
Definitions
Determinant of a Matrix Product
For matrices Anx n & Bnx n , of order “ n ”
det ( AB ) = det ( A ) det ( B )
|AB | = | A | | B |

Determinant of a Scalar Multiple of a Matrix
For matrix Anx n of order “ n ” , and Scalar “ c ”
det ( cA ) = cn det ( A )
|A | = cn | A |

Determinant of an Invertible Matrix
For matrix Anx n
A is invertible if-and-only-if
det ( A ) ≠ 0
|A | ≠ 0

Factors are not row-column specific
(for whatever reason??)
An invertible matrix must have a non-
zero determinant, elsewise one would be
dividing by zero to obtain the inverse of
the matrix (undefined)

© Art Traynor 2011
Mathematics
Linear Algebra
Definitions
Determinant of an Inverse Matrix
For matrix Anx n
A is invertible if-and-only-if
det ( A– 1 ) =
|A | =

1
det ( A )
1
|A |
Determinant of a Transpose
For matrix Anx n
det ( A ) = det ( AT )
|A | = |AT |

An invertible matrix must have a non-
zero determinant, elsewise one would be
dividing by zero to obtain the inverse of
the matrix (undefined)

© Art Traynor 2011
Mathematics
Linear Algebra
Definitions
Equivalent Conditions For A Non-Singular Matrix
For matrix Anx n , the following statements are equivalent
“ A ” is invertible
Ax = b has a unique solution for every n x 1 column matrix b
Ax = 0 has only the trivial solution
“ A ” is row-equivalent to In

“ A ” can be written as the product of elementary matrices
det ( A ) ≠ 0 ; |A | ≠ 0

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Adjoint of a Matrix

The Cofactor Cij
of the entry aij
 C11
C21
Cn1
.
.
.
C12
C22
Cn2
.
.
.
. . .
. . .
. . .
.
.
.
C1n
C2n
Cnn
.
.
.
Cofactor Matrix of A
C11
C12
C1n
.
.
.
C21
C22
C2n
.
.
.
. . .
. . .
. . .
.
.
.
Cn1
Cn2
Cnn
.
.
.
adj ( A ) =
Adjoint Matrix of A
The transpose of the Cofactor Matrix Cij
of “ A ” is Cij
T


© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Adjoint Equivalence with Matrix Inverse
For invertible matrix Anx n , A– 1 is defined by
A– 1 = adj ( A )
A– 1 = adj ( A )
 1
det ( A )
1
|A |

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Cramer’s Rule
Given a square matrix Anx n in “ n ” equations ( i.e. LE count = Airty )
a11 x1 + a12 x2 = b1
a21 x1 + a22 x2 = b2
b1 a22 – b2 a12
a11 a22 – a21 a12
x1 =
b2 a11 – b1 a21
a11 a22 – a21 a12
x2 =
which denominator forms the Determinant of the matrix “ A ”
b1
b2
a12
a22
x1 =
a11
a21
a12
a22
a11
a21
b1
b2
, x2 =
a11
a21
a12
a22
, a11 a22 – a21 a12 ≠ 0
|A1 | =
b1
b2
a12
a22
|A2 | =
a11
a21
b1
b2
|A1 |
|A |
x1 =
|A2 |
|A |
x2 =

© Art Traynor 2011
Mathematics
Algebra Of Matrices
Linear Algebra
Cramer’s Rule
Given a system of “ n ” linear equations
in “ n ” variables ( i.e. LE count = Airty )
with coefficient matrix “ A ”
and non-zero determinant |A |
the solution of the system is given as:

|A1 |
|A |
x1 = , x2 = , … , xn = where the ith column of Ai is the
“ constant ” vector in the LE system
|A2 |
|A |
|An |
|A |
Linear Equation System
a11 x1 + a12 x2 + a13 x3 = b1
a21 x1 + a22 x2 + a23 x3 = b2
a31 x1 + a32 x2 + a33 x3 = b3
Matrix-Vector Notation
Example:
a11 b1
a21 b2
a31 b3
a12
a22
a32
a11 a13
a21 a23
a31 a33
a12
a22
a32
|A3 |
|A |
x3 = =

© Art Traynor 2011
Mathematics
Topological
Spaces
Space
Mathematical Space
Mathematical Space
A Mathematical Space is a
Mathematical Object that is
regarded as a species of Set
characterized by:

Structure
Heirarchy
and Inner Product
Spaces
Normed
Vector Spaces
Vector Spaces
Metric Spaces
Subordinate Spaces ( Subspaces ) inherit
the properties of Parent Spaces
such that subordinate Subspaces are said to Induce their
properties onto the parent spaces in a recursive fashion
e.g. an Algebra or Algebraic Structure

© Art Traynor 2011
Mathematics
ProjectiveEuclidean
Mathematical Space
Distance between two
points is defined
Distance is Undefined
Space
Mathematical Space
Heirarchy
Upper Level Classification
Second Level Classification Non-EuclideanEuclidean
Finite Dimensional Infinite Dimensional
Compact Non-Compact
Second Level Classification N
n
, Z
n
, Q
n
, R
n
, C
n
, E
n
This slide is very slippery
It really needs a deeper dive
to achieve necessary cogency

© Art Traynor 2011
Mathematics
MapFunction Morphism
A Relation between a Set of
inputs and a Set of permissible
outputs whereby each input is
assigned to exactly one output
A Relation as a Function but
endowed with a specific
property of salience to a
particular Mathematical Space
A Relation as a Map with the
additional property of Structure
preservation as between the
sets of its operation
Structure
A Set attribute by which several species of Mathematical
Object are permitted to attach or relate to the Set
which expand the enrichment of the Set

Space
Mathematical Space
Measure
The manner by which a
Number or Set Element is
assigned to a Subset
Algebraic Structure
A Carrier Set defined by one or
more Finitary Operations
Field
A non-zero Commutative Ring
with Multiplicative Inverses for all
non-zero elements
(an Abelian Group under Multiplication)
 

© Art Traynor 2011
Mathematics
FMM
A unique Relation between Sets
Structure

Space
Mathematical Space
Measure
Algebraic Structure
Field
A non-zero Commutative Ring
with Multiplicative Inverses for all
non-zero elements
(an Abelian Group under Multiplication)
Satisfies Group Axioms plus Commutativity
Arithmetic Operations are defined ( +, – , x ,÷ )
Salient to a Mathematical Space
Preserving of Structure
FMM = Function~Map~Morphism
Akin to the Holy Trinity
Topology
Those properties of a
Mathematical Object
which are invariant under
Transformation or Equivalence
Metric Space
A Set for which distance between
all Elements of the Set are defined
The Triangle Inequality
constitutes the principle Axiom
from which three subsidiary
axioms are derived
F ≡ C R Q Z N

© Art Traynor 2011
Mathematics
Topology
Structure

Space
Mathematical Space
Manifold
A Topologic Space resembling a
Euclidean Space whose features
may be charted to Euclidean
Space by Map Projection
Metric Space
Riemann Manifold
Order
A Binary Set Relation exhibiting
the Reflexive, Antisymmetric,
and Transitive properties
Equivalence Class
Mathematical Object
Surface of a Sphere is not a
Euclidean Space!
A Real Manifold enriched with an
inner product on the Tangent Space
varying smoothly at each point
Geometry
A Complete, Locally Homogenous,
Reimann Manifold
Scale Invariant - Exhibits Multiplicative Scaling
Convergent
the Reflexive, Symmetric, and
Transitive properties

© Art Traynor 2011
Mathematics
Topology
Structure

Space
Mathematical Space
Manifold
A Topologic Space resembling a
Euclidean Space whose features
may be charted to Euclidean
Space by Map Projection
Order
the Reflexive, Antisymmetric,
and Transitive properties
Equivalence Class
Mathematical Object
Surface of a Sphere is not a
Euclidean Space!
the Reflexive, Symmetric, and
Transitive properties
Differential Structures
A Structure on a Set rendering the
Set into a Differential Manifold with
n-dimensional Continuity defined
by a CK Atlas of Bijection/Charts
Categories
Comprised of
Object and Morphism Classes
and Morphisms relating the Objects
admitting Composition
and satisfying the
Associativity and Identity Axioms

© Art Traynor 2011
Mathematics
FMM
A unique Relation between Sets
Structure

Space
Mathematical Space
Measure
Salient to a Mathematical Space
Preserving of Structure
FMM = Function~Map~Morphism
Akin to the Holy Trinity
SurjectionInjective
Functions
One-to-One Onto
Bijection
Inversive
One-to-One & Onto
f : X  Y
A Function which returns
a CoDomain equivalent to
the Domain of another
Function returning that
same CoDomain
aka: Automorphism

© Art Traynor 2011
Mathematics
Subspace
Subspace ( General )
Mathematical Space
Somewhat trivially, a mathematical Subspace
is a Subset
of a parent Mathematical Space
which inherits and enriches the Structure
of the superordinating Mathematical Space
 Closed under the operation of
Addition and Scalar Multiplication
And which satisfy the ten axioms
governing vector space elements
Mathematical
Space
M
Mathematical
Space
Given Mathematical Space “ M “①
Example:

© Art Traynor 2011
Mathematics
Subspace
Mathematical Space
is a Subset
Mathematical
Space
M
Mathematical
Space
Given Mathematical Space “ M ”①
Example:
Set Theory entails that at least two improper subspaces
are constituent of the Space: the Empty Set and “ M ” itself
Proof:
•Let M be a Space over some field F.
•Every Space must contain at least two elements:
the empty set { } , and itself Ms
P ( M )
Ms

© Art Traynor 2011
Mathematics
Subspace
Mathematical Space
is a Subset
Mathematical
Space
M
Mathematical
Space
Given Mathematical Space “ M ”①
Example:
Set Theory entails that at least two improper subspaces
are constituent of the Space: the Empty Set and “ M ” itself
P ( M )
Ss
② We next introduce into this Power Set / Spanning space
a well defined non-zero element “ S ” and at least one
additional Structuring operation (∗) such that
Si ∗ Si Fi  Ss
Si
∗

© Art Traynor 2011
Mathematics
Vector Space
Vector Space ( General )
Mathematical Space
Vector Spaces
Vector
Spaces
Metric
Spaces
Topological
Spaces
A Vector Space V
is a species of Set over a Field F of scalars (e.g. R or C )
whose constituent point elements can be uniquely characterized by
an ordered tuple of n-dimension ( Vectors )
Structured the Superposition Principle ( and its derivative
Linear Operations ):

Addition ( aka Additivity Property )
A function that assigns to the combination of any two or more elements
of the space a resultant unique n-tuple ( Vector ) composed of the
sum of the respective operand vector components.
f ( 〈 a , b 〉 ) = 〈 an + bn 〉 = 〈 rn 〉 = r ( e.g. rn  R n
)
f : V + V  V = a  V  f ( a )  V
Closed under the operation of
The simplest Vector Space is the
space populated by only the Field
itself, known as a Coordinate Space
Vector Addition corresponds to the Motion of Translation

© Art Traynor 2011
Mathematics
Addition
Vectors
Vector (General )
x
y
O initial point
terminal point
Free Vector
r
A
 Sum of Vectors – Vector Addition (Tail –to–Tip)
B
a ( ax, ay )
b ( bx, by )
║a ║
║b ║
 Any two (or more) vectors can be summed by positioning the operand
vector (or its corresponding-equivalent vector) tail at the tip of the
augend vector.
 The summation (resultant) vector is then extended from (tail) the
origin (tail) of the augend vector to the terminal point (tip) of the
operand vector (tip-to-tip/head-to-head).
ry
rx
r ( rx , ry )
θ
“ Tail-to-Tip ”
“ Tip-to-Tip ”
Same procedure, sequence of
operations whether for vector
addition (summation) or vector
subtraction (difference)
Resultant is always tip-to-tip
Operands are oriented “ tip-to-tail ”
resultant vector is oriented “ tip-to-
tip ”
The resultant vector in a
summation always originates at
the displacement origin and
terminates coincident at the
terminus of the final displacement
vector (e.g. tip-to-tip)
Chump Alert: A vector summation
is a species of Linear Comination

© Art Traynor 2011
Mathematics
Vector Space
Mathematical Space
Vector Spaces
Vector
Spaces
Metric
Spaces
Topological
Spaces
A Vector Space V
is a species of Set
over a Field F of scalars (e.g. R or C )
whose constituent point elements can be uniquely characterized by
an ordered tuple of n-dimension ( Vectors )
Structured by the following Linear Operations:

Addition
f ( c 〈 an 〉 ) = 〈 can 〉 = 〈 rn 〉 = r ( e.g. rn  R n
)
A function that assigns to the combination of any element of the
multiplicand field and any multiplier vector space element a resultant
unique n-tuple ( Vector ) composed of the product of the respective
multiplicand scalar and the constituent multiplier vector n-components
supra.
f : F x V  V = a  V  f ( a )  V

© Art Traynor 2011
Mathematics
Vectors
Vector (General )
PQ
x
y Position Vector
O initial point
Free Vector
C
 Vector Scalar Multiple – a species of Transformation where the
CoDomain Set if positive effects an Expansion,
or if negative effects a Dilation.

O
C ( cax , cay )
A ( ax , ay )
c OA = OC
terminal point
Example: F = ma
Vector Scalar Multiple
with the multiplicand ( vector to be
scaled ) “ scaled ” by the
multiplier-scalar.
The result constitutes a vector
addition of the product of the
scalar and the multiplicand
normalized unit vector (NUV) thus
preserving multiplicand orientation
in the result
c 〈 ax , ay 〉 = 〈 cax , cay 〉
Chump Alert: A vector scalar is a
species of Linear Comination

© Art Traynor 2011
Mathematics
Vector Space
Mathematical Space
Vector Spaces
Vector
Spaces
Metric
Spaces
Topological
Spaces
A Vector Space V is a species of Set over a Field F
of scalars (e.g. R or C ) whose constituent point elements can be
uniquely characterized by an ordered tuple of n-dimension ( Vectors )

Addition
Vector/Linear (VL ) Spaces are said to be “Algebraic”
VL Space operations define figures (subspaces?) such as lines and planes
The Dimension of a VL Space is determined by the maximal
number of Linear Independent variables (identical to the minimal
number of vectors that Span the space)
Additional Structure apart from that characterizing general Vector
Space is needed to define Nearness, Angles, or Distance
A Vector Space V is a species of Set over a Field F

© Art Traynor 2011
Mathematics
Vector Space
Mathematical Space
Vector Spaces
Vector
Spaces
Metric
Spaces
Topological
Spaces

Addition
Vector Spaces are said to be “Linear” spaces
(as distinct from Topological Spaces)
Vector/Linear (VL ) Spaces are said to be “Algebraic”
VL Space operations define figures (subspaces?) such as lines and planes
The Dimension of a VL Space is determined by the maximal
number of Linear Independent variables (identical to the minimal
number of vectors that Span the space)
Additional Structure apart from that characterizing general Vector
Space is needed to define Nearness, Angles, or Distance

© Art Traynor 2011
Mathematics
Vector Space
Mathematical Space
Vector Spaces
Vector
Spaces
Metric
Spaces
Topological
Spaces

Addition
The essential Structure of a Vector Space enables transformations of
its elements that correspond to classes of Motion
Vector Addition (as well as Scalar Multiplication, which is by extension
repeated vector addition) corresponds to Translation
Translation is classed as one of three species of Rigid Motion the
other two Rotation and Reflection require additional Structure
A Vector is understood to represent a difference (Displacement )
between the respective values of its constituent ordered tuples

© Art Traynor 2011
Mathematics
Vectors
Vector Properties
Multiplicative Inverse
If c( v ) = 0
Zero Vector
Scalar Identity
– 1( v ) = – v
Properties of Scalar Multiplication
If v represents any element of a vector space V ( v  V )
and c represents any scalar, then the following properties pertain:

Zero Vector
Multiplicative Identity0( v ) = 0
Scalar Zero Vector
c( 0 ) = 0
then c = 0
or v = 0

© Art Traynor 2011
Mathematics
Vector Space Axioms – Addition Abstraction
Mathematical Space
Vector Space
A vector space is comprised of four elements: a set of vectors,
a set of scalars, and two operations:

u + v ( is in V ) Closure Under Addition
( u + v ) + w = u + ( v + w )
Changes Order of Operations
as per “PEM-DAS”, Parentheses
are the principal or first operation
u + v = v + u
Commutative Property
of Addition
Re-Orders Terms
Does Not Change
Order of Operations – PEM-DAS
Associative Property
of Addition
u + 0 = u Additive Identity
u + ( – u ) = 0 Additive Inverse
If V is a vector space
then $ 0 | "u  V
" u  V ,
$ – u | "u  V
Operations: Addition & Scalar Mult.
Represents “ 0 = – 2 ” ,
a contradiction,
and thus no solution {  }
to the LE system for which
the augmented matrix stands
Note that there is nothing
in these axioms that entails
Length/Distance or Magnitude of Vectors,
nor corresponding attributes
such as Angle or Nearness

© Art Traynor 2011
Mathematics
cu ( is in V )
Closure Under
c ( u + v ) = cu + cv Distributive
( c + d )u = cu + du Distributive
c( du ) = ( cd )u
Associative Property
of Multiplication
1( u ) = u
Vector Space Axioms – Scalar Multiplication Abstraction
A vector space is comprised of four elements: a set of vectors,
a set of scalars, and two operations:

Operations: Addition & Scalar Mult.
Let c = 0 and you therefore don’t need
to state a separate scalar multiplicative
zero element
Mathematical Space
Vector Space
Note that there is nothing
in these axioms that entails
Length/Distance or Magnitude of Vectors,
nor corresponding attributes
such as Angle or Nearness

© Art Traynor 2011
Mathematics
Normed Vector Space
Normed Vector Space
Mathematical Space
“Length” (aka Distance/Magnitude) is
one type of norm
The vector norm ║X║ is more formally
defined as the ℓ2-norm
Somewhat trivially, a Normed Vector Space
is a Vector Space
Structured by an Norm
 A norm is defined as a Mathematical
Structure of the species of a Measure.
Inner Product
Spaces
Normed
Vector Spaces
Vector Spaces
There are several species of Norm
A Norm on a vector space V is a function
that maps a vector a in V to an element r of a Field F
f : V R = a  V  f ( a )  F = rn ( e.g rn  R n
)
2 2
ax + ay║a ║ = ║〈 ax , ay 〉║ =Magnitude
p-Norm
aka Euclidean Norm, L2 Norm, or ℓ2Norm ( or “ Length ” )
║a ║p = ( Σ | ai |p
)i = 1
n 1
p

© Art Traynor 2011
Mathematics
Magnitude
Vectors
Vector ( General ) Aka: Geometric or Spatial Vector
From the Latin Vehere (to carry)
x
y
O
θ
A ( ax , ay )
 Magnitude
a
Position Vector
PVF: Position Vector Form
xO
θ
a (adj )
b (opp )
r = c (hyp )
M A (1, 0)
P ( cos θ, sin θ )
1
tan θ
cos θ
Q
sin θ
y
UCF: Unit Circle Form
ay
ax
Unit Circle
In PVF the magnitude of a vector a = 〈 ax , ay 〉 is equivalent to the
hypotenuse ( c = ║a ║ ) of a right triangle whose adjacent side
( a ) is given by the coordinate a1 , and whose opposite side ( b )
is given by the coordinate a2 :
2 2
ax + ay║a ║ = ║ 〈 ax , ay 〉 ║ =
Pythagorean Theorem derived

© Art Traynor 2011
Mathematics
Inner Product Space
Inner Product Space ( IPS )
An Inner Product Space
is a Vector Space
over a Field of Scalars (e.g. R or C )
Structured by an Inner Product

Inner Product
Spaces
Normed
Vector Spaces
For a Euclidean Space the Inner
Product is defined as the Dot
Product
Positive-Definite Symmetric
Bilinear Form
Length/Distance or Magnitude of Vectors
This Structure moreover defines IPS salients such as:
Vector Subtended Angle
Orthogonality of Vectors
These spaces have a well-ordered semantic construction of the form: A Vector Space with an Inner
Product “on” it…
“ The IPS of conventional multiplication over the field of R ”
“ The IPS of the dot product over the field of R ”
Mathematical Space

© Art Traynor 2011
Mathematics
Inner Product Space ( IPS ) Axioms
Inner Product
Spaces
Normed
Vector Spaces
A summation
of the Scalar Product
of the vector components, a = 〈 ax , ay 〉 , b = 〈 bx , by 〉
For a Euclidean Space the Inner
Product is defined as the Dot
Product
Note here the possibility of
describing the dot product as an
equivalence class for its alternate
expressions

to every n-tuple of vectors a and b in V, a scalar in F
Let V be a vector space over F , a Field of Scalars (e.g. R or C ) .
An Inner Product on V is a function that assigns,
f ( 〈 a , b 〉 ) = ( an bn + an bn ) = rn ( e.g. rn  R n
)
Inner Product Space
Mathematical Space

© Art Traynor 2011
Mathematics
Inner Product Axioms
Given vectors u , v , and w in Rn , and scalars c , the following axioms pertain:
〈 u , v 〉 = 〈 v , u 〉 Symmetry
〈 u , v + w 〉 = 〈 u , v 〉 + 〈 u , w 〉 Additive Linearity
Positive Definiteness
Inner Product Space
Mathematical Space
c 〈 u , v 〉 = 〈 cu , v 〉 Multiplicative Linearity
〈 v , v 〉 ≥ 0 〈 v , u 〉
〈 v , v 〉 = 0
if and only if v = 0

© Art Traynor 2011
Mathematics
Vectors
x
y Position Vector
O initial point
terminal point
Free Vector
A
B
O
A ( ax , ay )
B ( bx , by )
║a ║
║b ║
Vector ( Euclidean )
 Dot Product
The dot product of two vectors
is the scalar summation
of the product
Aka: Geometric or Spatial Vector
PVF: Position Vector Form
UCF: Unit Circle Form
a · b = ax bx + ay by
of their components, a = < ax , ay > , b = < bx , by >
Also referred to as the Scalar Product or Inner Product Pythagorean Theorem derived
Inner (Dot) Product

© Art Traynor 2011
Mathematics
Vectors
x
y Position Vector
O initial point
terminal point
Free Vector
A
B
O
A ( ax , ay )
B ( bx , by )
║a ║
║b ║
 Dot Product & Angle Between Vectors
For any two non-zero vectors sharing a common initial point
the dot product of the two vectors is equivalent to
the product of their magnitudes and the cosine of the angle between
Inner (Dot) Product
θ θ
a · b = ║b ║║a ║ cosθ
cosθ =
║a ║║b ║
a · b
You will be asked to find the angle
between two vectors sharing a
common initial point (origin)…a lot
θ = cos– 1
║a ║║b ║
a · b

© Art Traynor 2011
Mathematics
Vectors
x
y Position Vector
O
Free Vector
Physical Quantities represented
by vectors include: Displacement,
Velocity, Acceleration, Momentum,
Gravity, etc.
O
A ( ax , ay )
B ( bx , by )
a
b
c
A
B
θ θ
║a ║ cos θ
a · b = ║b ║║a ║ cosθ
= a · b
OB – OA = AB
Inner (Dot) Product

© Art Traynor 2011
Mathematics
Vectors
x
y
Position Vector
O
Free Vector
Gravity, etc.
O
A ( a1, a2 )
B ( b1x , by )
a
b
c
A
B
θ θ
║a ║ cos θ
║b ║
Area = ║b ║║a ║ cosθ
= a · b
OB – OA = AB
Inner (Dot) Product

© Art Traynor 2011
Mathematics
Vectors
x
y
Position Vector
O
A ( ax , ay )
B ( bx , by )
a
b
θ
 Vector Component as Projection
Inner (Dot) Product
②
③
④
The intersection of any two vectors with
common origin will feature a shared angle
( the “Angle Between” ).
①
② In Position Vector Form (PVF), the vector
system can be aligned so that the vector
common origin coincides with a coordinate
system origin and one of the vectors (the
Multiplier vector “ b ”) can then be aligned
along the x-coordinate axis
③ In this orientation the Multiplicand vector
“ a ” (if the angle between is acute) will
terminate in the first quadrant of the
coordinate system.
O
A
B
θ
Free Vector
①
④ Note that the X-component of a (i.e. ax) is
geometrically equivalent to a vertical
projection from a onto the X-axis and b
ax

© Art Traynor 2011
Mathematics
Vectors
x
y
Position Vector
O
A ( ax , ay )
B ( bx , by )
a
b
θ
Inner (Dot) Product
⑤ Recalling the trigonometric relationships
of the Unit Circle, it can be further noted
that the X-component of a (i.e. ax) –
previously noted to be geometrically
equivalent to a vertical projection from a
onto the X-axis and b – is also
geometrically equivalent to the product of
the length of a (its Magnitude) and the
cosine of the angle formed with the x-axis
ax
2 2
ax + ay║a ║ = ║〈 ax , ay 〉║ =
ax = ║a ║ cosθ
║b ║
1
compb a = a · b
O
U
!
!
cos θ
r = 1 = c
r = c (hyp )
tan θ
sin θ
θ
x
y
a (adj )
b (opp )
Unit Circle

© Art Traynor 2011
Mathematics
Vectors
x
y
Position Vector
O
A ( ax , ay )
B ( bx , by )
a
b
θ
Inner (Dot) Product
⑤ Recalling the trigonometric relationships
of the Unit Circle, it can be further noted
that the X-component of a (i.e. ax) –
previously noted to be geometrically
equivalent to a vertical projection from a
onto the X-axis and b – is also
geometrically equivalent to the product of
the length of a (its Magnitude) and the
cosine of the angle formed with the x-axis
ax
2 2
ax + ay║a ║ = ║〈 ax , ay 〉║ =
Another way to express this geometrical
equivalence is to note the inherent
relationship between the lengths of two
vectors in composition sharing a common
origin and the angle between the two
supplied by the inner (dot) product
relationship
xO
θ
a (adj )
b (opp )
r = c (hyp )
M A (1, 0)
P ( cos θ, sin θ )
1
tan θ
(r|c|1) cos θ
Q
sin θ
y
Unit Circle
a
OA = OP = AP = t = θ = 1

© Art Traynor 2011
Mathematics
Vectors
x
y
Position Vector
O
Gravity, etc.
A ( a1, a2 )
B ( b1x , by )
a
b
c
θ
║a ║ cos θ
║b ║
Area = ║b ║║a ║ cosθ
Inner (Dot) Product
a · b = ║b ║║a ║ cosθ

© Art Traynor 2011
Mathematics
Vectors
xO
The notion of “component along” is
a direct consequence of the
definition of an inner (dot) product –
relating the two “sides” of a vector
“triangle” via a ratio given by the
cosine of the “angle between”
O
A ( ax , ay )
B ( b1, b2 )
a
b
c
A
Bθ θ
║a ║ cos θ
 Vector Component Along an Adjoining Vector
y
Position Vector
The component of OA along OB
that has the same direction as OB
║b ║
1
compb a = a · b
x
y
Position Vector
║b ║
b
compb a = a ·
Compb a = a ·║b ║
b
is the dot product of OA with the unit vector 1
║u ║
u
║a ║
û = u =( (
Dot Product

© Art Traynor 2011
Mathematics
Vectors
Col. 1 Col. 2 Col. 3 . . . Col. n
a1 a2 a3 . . . ana · b = aTb =
Col. 1
b1
b2
b3
bm
.
.
.
 Dot Product ( Determinant Form )
The dot product of two vectors, is the matrix product
of the 1 x n transpose of the multiplicand vector
and the m x 1 multiplier vector
Col. 1
b1
b =
b2
b3
bm
.
.
.
Col. 1
a1
a =
a2
a3
an
.
.
.
a · b = aTb = a1b1 + a2b2 + a3b3 …+ anbm
Inner (Dot) Product

© Art Traynor 2011
Mathematics
Vector Spaces – Rn
Vectors
Tuple Properties
An “ n-tuple” is characterized by the following:
A sequence
An ordered list
Comprising “ n ” elements ( n is a non-negative integer)
Canonical “ n-tuples ”
0-tuple: null tuple
1-tuple: singleton
2-tuple: ordered pair
3-tuple: triplet

© Art Traynor 2011
Mathematics
Vectors
Vector “ n-tuple ” Representation
An ordered n-tuple represents a vector in n-space
n Of the form ( ai , ai+1 ,…an – 1 , an )
n The Set of all n-tuples is n-space, denoted by Rn
n An n-tuple can be rendered as a point in Rn whose coordinates
describe a unique vector a
n n-tuples delineate Rn such that all points in Rn can be
represented by a unique n-tuple
Tuple Properties

© Art Traynor 2011
Mathematics
Vectors
“ n-tuple ” distinguished from a set
n-tuples delineate Rn space such that tuples
of disparate n-order: ( 1, 2, 3, 2 ) ≠ ( 1, 2, 3 )
are not equal as the same sequence expressed
as elements of a set { 1, 2, 3, 2 } = { 1, 2, 3 }

Tuple elements are ordered: ( 1, 2, 3 ) ≠ ( 3, 2, 1 )
whereas for a set { 1, 2, 3 } = { 3, 2, 1 }

A tuple is composed of a finite population of elements
whereas a set may contain infinitely many elements

Tuple: Sequence Matters
Set: Sequence Does Not Matter
Set: Order Matters
Set: Order Does Not Matter
Tuple Properties

© Art Traynor 2011
Mathematics
Vectors
Tuples as Functions
An n-tuple can be rendered as a function “ F ”
the domain of which is represented by the tuple’s element index/indices or “ X ”
the codomain of which is represented by the tuple’s elements or “ Y ”
X = { i , i + 1 ,…, n – 1 , n }
( ai , ai+1 ,…an – 1 , an ) = ( X , Y , F )
( a1 , a2 ,…, an – 1 , an ) = ( X , Y , F )
X = { 1 , 2 ,…, n – 1 , n }
or
or
Y = { a1 , a2 ,…, an –1 , an }
F = { ( 1, a1 ) , ( 2, a2 ) ,…, ( n – 1, an – 1 ) , ( n, an ) }
Tuple Properties

© Art Traynor 2011
Mathematics
Definition
Vectors
A geometric object (directed line segment)
describing a physical quantity and characterized by
Direction: depending on the coordinate system used to describe it; and
Magnitude: a scalar quantity (i.e. the “length” of the vector)
originating at an initial point [ an ordered pair : ( 0, 0 ) ]
and concluding at a terminal point [ an ordered pair : ( a1 , a2 ) ]
Other mathematical objects
describing physical quantities and
coordinate system transforms
include: Pseudovectors and
Tensors
 Not to be confused with elements of Vector Space (as in Linear Algebra)
 Fixed-size, ordered collections
 Aka: Inner Product Space
 Also distinguished from statistical concept of a Random Vector
or from Vectus…to carry some-
thing from the origin to the point
constituting the components of the vector 〈 a1 , a2 〉

© Art Traynor 2011
Mathematics
Vectors
Vector ( Euclidean ) Aka: Geometric or Spatial Vector
 Vector – Properties ( PVF Form)
Each (position) vector determines a unique Ordered Pair ( a1 , a2 )
The coordinates a1 and a2 form
the Components of vector 〈 a1 , a2 〉
x
y Position Vector
║a ║
O
θ
A ( a1, a2 )
initial point
terminal point
a
a1
a2


Position Vector
 A vector represented in PVF is Unique
n There is precisely one free-vector equivalent in PVF: a = OA
n The unique ordered pair describing the vector
is a unique n-tuple in Rn

© Art Traynor 2011
Mathematics
Vector Standard Operations in Rn
Vectors
Sum of Vectors: u + v
u + v = ( u1+ v1 , u2+ v2 , … , un –1 + vn –1 , un + vn )
Given u = ( ui , ui+1 ,…un – 1 , un ) and
v = ( vi , vi+1 ,…vn – 1 , vn )
Scalar Multiple of Vectors: cu
cu = ( cu1 , cu2 , … , cun –1 , cun )
Vector Operations

© Art Traynor 2011
Mathematics
Addition
Vectors
x
y
O initial point
terminal point
Free Vector
r
A
 Sum of Vectors – Vector Addition (Tail –to–Tip)
B
a ( ax, ay )
b ( bx, by )
║a ║
║b ║
 Any two (or more) vectors can be summed by positioning the operand
vector (or its corresponding-equivalent vector) tail at the tip of the
augend vector.
 The summation (resultant) vector is then extended from (tail) the
origin (tail) of the augend vector to the terminal point (tip) of the
operand vector (tip-to-tip/head-to-head).
ry
rx
r ( rx , ry )
θ
“ Tail-to-Tip ”
“ Tip-to-Tip ”
tip ”
The resultant vector in a
summation always originates at
the displacement origin and
terminates coincident at the
terminus of the final displacement
vector (e.g. tip-to-tip)
Chump Alert: A vector summation

© Art Traynor 2011
Mathematics
Vector Standard Operations in Rn
Vectors
Vector Operations
Difference of Vectors: u – v
u – v = ( u1 – v1 , u2 – v2 , … , un –1 – vn –1 , un – vn )
Given u = ( ui , ui+1 ,…un – 1 , un ) and
v = ( vi , vi+1 ,…vn – 1 , vn )
Scalar Multiplicative Inverse: – cu ( c = 1)
– u = ( – u1 , – u2 , … , – un –1 , – un )

© Art Traynor 2011
Mathematics
Subtraction
Vectors
x
y
O
ry
rx
θ
initial
point
terminal point
Free Vector
r = a + bcorr
a
b
O
“ Tail-to-Tip ”
“ Tip-to-Tip ”
( Addition )
bcorr
– bcorr “ Tip-to-Tip ”
( Difference )
Position Vector
r = a – bcorr
Difference of Vectors – Vector Subtraction ( Tail –to–Tip )
 Any two (or more) vectors can be subtracted by positioning the tail of
a corresponding-equivalent subtrahend vector (initial point) at the
tip (terminal point) of the minuend vector.
 The difference (resultant) vector is then extended from the tail (initial
point ) of the minuend vector (tail-to-tail) to the terminal point
(tip) of the subtrahend vector (tip-to-tip).
minuend
subtrahend
r
r = a + bcorr
bcorr
– bcorr
a
b
tip ”
Chump Alert: A vector difference

© Art Traynor 2011
Mathematics
Vector Properties – Additive Identity & Additive Inverse
Vectors
Vector Properties
Given vectors u , v , and w in Rn , and scalars c and d, the following
properties pertain

0v = 0 Scalar Zero Element
If u + v = v then u = 0
Additive Identity
is Unique
If v + u = 0 then u = – v Additive Inverse
is Unique
c0 = 0
Scalar Multiplicative Identity
of Zero Vector
If cv = 0 then c = 0 or v = 0 Zero Vector
Product Equivalence
– ( – v ) = v Negation Identity

© Art Traynor 2011
Mathematics
Vector Spaces – Classification
Real Number Vector Spaces
R = set of all real numbers
R2 = set of all ordered pairs
R3 = set of all ordered triplets
Rn = set of all n-tuple
Matrix Vector Spaces
Mm,n = set of all m x n matrices
Mn,n = set of all n x n square matrices
Vector Spaces
Vector Space

© Art Traynor 2011
Mathematics
Vector Spaces
Vector Spaces – Classification
Polynomial Vector Spaces
P = set of all polynomials
Pn = set of all polynomials of degree ≤ n
Continuous Functions ( Calculus ) Vector Spaces
C ( – ∞ , ∞) = set of all continuous functions
defined on the real number line

C [ a, b ] = set of all continuous functions
defined on a closed interval [ a, b ]

Vector Space

© Art Traynor 2011
Mathematics
Vector Subspaces
Subspace Definition
A non-empty subset W ( W ≠  ) of a vector space V
is a subspace of V when the following conditions pertain:

W is a vector space under addition in V
W is a vector space under scalar multiplication in V
Subspace Test
For a non-empty subset W ( W ≠  ) of a vector space V,
W is a subspace of V if-and-only if the following pertain:

If u and v are in W, then u + v is in W
If u is in W and c is any scalar, then cu is in W
Zero Subspace
W = { 0 }
Vector Space

© Art Traynor 2011
Mathematics
Vector Subspaces
W 1
Polynomial
functions
W 5
Functions
W 2
Differentiable
functions
W 3
Continuous
functions
W 4
Integrable
functions
W5 = Vector Space " f Defined on [ 0, 1 ]
W4 = Set " f Integrable on [ 0, 1 ]
W3 = Set " f Continuous on [ 0, 1 ]
W2 = Set " f Differentiable on [ 0, 1 ]
W2 = Set " Polynomials
Defined on [ 0, 1 ]
W1  W2  W3  W4  W5
W 1 – Every Polynomial function is Differentiable W1  W2
W 2 – Every Differentiable function is Continuous W2  W3
W 3 – Every Continuous function is Integrable W3  W4
W 4 – Every Integrable function
is a Function W4  W5
Function Space Section 4.3, (Pg. 164)
Vector Space

© Art Traynor 2011
Mathematics
Vector Subspaces
U
V W
V  W
Properties of Scalar Multiplication
If V & W are both subspaces of a vector space U,
then the intersection of V & W ( V  W )
is also a subspace of U

Vector Space

LinearAlgebra_160423_01

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