This document discusses inner product spaces and properties of the inner product. It provides examples of determining the inner product of vectors and applying properties like commutativity, distributivity, and associativity. It also defines length in Rn and discusses the Euclidean plane E2, defining distance between points as the absolute value of their difference. Students will learn to determine if a function defines an inner product, find inner products of vectors, and solve for distances in E2.
5. At the end of the lesson the students will be able to:
Determine whether a function defines an inner
product.
Find the inner product of two vector in 𝑅𝑛.
OBJECTIVES
6. INNER-PRODUCT SPACE
It is represented by angle
brackets <u, v>
It is a vector space with additional
structure called an inner product.
It is a function that associate a
real number <u, v> that satisfies
the following axioms…
7. PROPERTIES OF INNER PRODUCT
Commutative Property of the Inner Product ( <u, v> = (v, u> )
Distributive Property of the Inner Product (<u, v+w>=<u, v>+<u, w>)
Associative Property of Inner Product ( k<u, v>=<ku, v>)
Positive Semi-Definite < 𝑣, 𝑣 > ≥ 0 if and only if v = 0
Point-Separate/Non-Degenerate < 𝑣, 𝑣 > = 0 if and only if v = 0
18. At the end of the lesson the students will be able to:
Define Euclidean plane.
Solve the distance between two vectors.
OBJECTIVES
19. THE EUCLIDEAN PLANE 𝑬𝟐
The plane has both algebraic and geometric
aspects. The algebraic properties focuses on
the vector properties of 𝑅2.
In the geometric properties we will focus on the
concept of Distance.
20. THE EUCLIDEAN PLANE 𝑬𝟐
If P and Q are points, we define the
distance between P and Q by the equation:
d(P, Q) = |𝑸 − 𝑷|
The symbol 𝑬𝟐
will be used to denote the
set of points in Euclidean plane equipped
with the distance function d.
22. THE EUCLIDEAN PLANE 𝑬𝟐
𝒅 𝑷, 𝑸 = |𝑸 − 𝑷|
Most important properties of the distance:
Theorem 5. let P, Q and R be points of 𝑬𝟐, 𝒕𝒉𝒆𝒏
i. d(P,Q)≥ 𝟎
ii. d(P,Q)=0 if and only if P=Q
iii. d(P,Q)=d(Q,P)
iv. d(P,Q)+d(Q,R)≥ 𝒅(𝑷, 𝑹)