Section 5.2: "Inner Product Spaces"

Definitions of Length, Distance, and Angle

Let u and v be any vectors in an inner product space V. 1. The length (or norm) of u is ||u|| = sqrt(<u,u>); 2. The distance between u and v is d(u,v) = ||u-v||; 3. The angle b/w 2 non0 vectors u and v is given by cosine(theta) = (<u,v>)/(||u||||v||), theta b/w 0 and pi 4. u and v are orthogonal when <u,v>=0;

THEOREM 5.9 Orthogonal Projection and Distance

Let u and v be two vectors in an inner product space V, such that v!= 0 vector. Then d(u,projvu) less than d(u,cv), where c != <u,v>/<v,v>.

THEOREM 5.8

Let u and v be vectors in an inner product space V 1. Cauchy-Schwarz Inequality:|<u,v>| less than or equal to ||u||||v||; 2. Triangle inequality: ||u+v|| less than or equal to ||u|| + ||v||; 3. Pythagorean Theorem: u and v are orthogonal if and only if ||u+v||^2 = ||u||^2 + ||v||^2;

Definition of Orthogonal Projection

Let u and v be vectors in an inner product space V, such that v != 0 vector. Then the orthogonal projection of u onto v is given by projvu = (<u,v>/<v,v>)*v.

Definition of Inner Product

Let u, v, and w be vectors in a vector space V, and let c be any scalar. An inner product on V is a function that associates a real number <u,v> with each pair of vectors u and v that satisfies the following axioms. 1. <u,v> = <v,u>; 2. <u, v+ w> = <u,v> + <u,w>; 3. c<u,v> = <cu,v>; 4. <v,v> is greater than or equal to 0, and <v,v> = 0 if and only if v = 0 vector.

THEOREM 5.7 Properties of Inner Products

Let u, v, and w be vectors in an inner product space V, and let c be any real number. 1. <0,v> = <v,0> = 0; 2. <u+v,w> = <u,w> + <v,w>; 3. <u,cv> = c<u,v>;