MATH 337-102 Lecture 13 - Sections 6.4 & 6.7

1. 6
6.1
© 2012 Pearson Education, Inc.
6.4 Gram-Schmidt

2. Slide 6.1- 2© 2012 Pearson Education, Inc.
Orthogonal Basis
Example 1: W = Span{x1, x2}.
𝐱1 =
3
6
0
, 𝐱2 =
1
2
2
Find an orthogonal basis for W.

3. Slide 6.1- 3© 2012 Pearson Education, Inc.
Gram-Schmidt
Algorithm to find an orthogonal basis, given a basis
1. Let first vector in orthogonal basis be first vector
in original basis
2. Next vector in orthogonal basis is component of
next vector in original basis orthogonal to the
previously found vectors.
Next vector less the projection of that vector onto
subspace defined by the set of vectors in the orthogonal
set
Scaling may be convenient
3. Repeat step 2 for all other vectors in original basis

4. Gram-Schmidt - Example
Example 2: W = Span{x1, x2, x3}.
𝐱1 =
1
1
1
1
, 𝐱2 =
0
1
1
1
, 𝐱3 =
0
0
1
1
Find an orthogonal basis for W.
Slide 6.1- 4© 2012 Pearson Education, Inc.

5. Orthonormal Basis
 All vectors have length 1
 Normalize after find orthogonal basis

6. Slide 6.1- 6© 2012 Pearson Education, Inc.
QR Factorization
Theorem 6-12: If A is mxn matrix with linearly
independent columns, then A can be factored as
A=QR, where Q is an mxn matrix whose columns form
an orthonormal basis for Col(A) and R is an nxn upper
triangular invertible matrix w positive entries on the
diagonal.
R = IR
=(QTQ)R, QTQ = I, since Q has orthonormal cols
= QT(QR)
= QTA

7. QR Factorization - Example
Find QR factorization of
𝐴 =
1 0 0
1 1 0
1 1 1
1 1 1
Slide 6.1- 7© 2012 Pearson Education, Inc.

8. 6
6.1
© 2012 Pearson Education, Inc.
6.7 Inner Product Spaces

9. Inner Product - Definition
Definition: An inner product on a vector
space V is a function that to each pair of
vectors u and v in V, associates a real
number <u,v> and satisfies the following
axioms for all u, v, w in V and all scalars c:
1. <u,v> = <v,u>
2. <u+v,w> = <u,w> + <v,w>
3. <cu,v> = c<u,v>
4. <u,u> ≥ 0 & <u,u>=0 iff u=0

10. Inner Product Space
 A vector space with an inner product is
called an inner product space.
 Example - Rn with the dot product is an
inner product space
Slide 6.1- 10© 2012 Pearson Education, Inc.

11. Inner Product - Example
u & v in R2, u = (u1, u2), v=(v1, v2)
Show <u,v> = 4u1u2 + 5v1v2 defines an inner
product
Slide 6.1- 11© 2012 Pearson Education, Inc.

12. Inner Product - Example
 V = P2 with inner product:
 <p,q> = p(0)q(0) + p(½)q(½) + p(1)q(1)
 p(t) = 12t2, q(t) = 2t-1
 <p,q> = ?
 <q,q> = ?
Slide 6.1- 12© 2012 Pearson Education, Inc.

13. Length, Distance, Orthogonality
 Length or norm:
||v|| = 𝐯, 𝐯
 Distance between u and v:
 Dist(u,v) = ||u-v||
 Orthogonal if <u,v> = 0
Slide 6.1- 13© 2012 Pearson Education, Inc.

14. Example
 ||p(t)|| and ||q(t)|| from previous example
Slide 6.1- 14© 2012 Pearson Education, Inc.

15. Gram-Schmidt
Let inner product be:
<p,q> = p(-2)q(-2) + p(-1)q(-1) + p(0)q(0) + p(1)q(1) + p(2)q(2)
Produce orthogonal basis for P2 by applying
Gram-Schmidt to: 1, t, t2
Slide 6.1- 15© 2012 Pearson Education, Inc.

16. Inequalities
 Cauchy Schwartz Inequality:
|<u,v>| ≥ ||u|| ||v||
 Triangle Inequality
||u+v|| ≤ ||u|| + ||v||
Slide 6.1- 16© 2012 Pearson Education, Inc.