Module 6 Linear Algebra: Definitions and Theorems

An inner product of a vector space V over real numbers is a function that associates a real number with every pair of vectors u and v in V in such a way that the following axioms are satisfied for all vectors u, v, and w in V and all scalars k

1. 1. <u,v> = <v,u> (symmetry axiom)

2. 2. <u+v,w> = <u,w> + <v,w> (additive axiom)

3. 3. <ku, v> = k <u, v> (homogeneity axiom)

4. 4. <v,v> >= 0 where <v,v> = 0 if and only if v = 0 (positivity axiom)

• True

• False

<u,v> = <v,u>

• Symmetry Axiom

• Additive Axiom

• Homogeneity Axiom

• Positivity Axiom

1. <u+v,w> = <u,w> + <v,w>

• Symmetry Axiom

• Additive Axiom

• Homogeneity Axiom

• Positivity Axiom

1. <ku, v> = k <u, v>

• Symmetry Axiom

• Additive Axiom

• Homogeneity Axiom

• Positivity Axiom

1. <v,v> >= 0 where <v,v> = 0 if and only if v = 0

• Symmetry Axiom

• Additive Axiom

• Homogeneity Axiom

• Positivity Axiom

• True

• False

• True

• False

• True

• False

• True

• False

Two vectors a, b in an inner product space V are called orthogonal if and only if <a,b> = 0

• True

• False