Orthogonal Basis

.

Find an orthogonal basis for W.

 

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

1. Repeat step 2 for all other vectors in original basis

Gram-Schmidt - Example

.

Find an orthogonal basis for W.

 

Orthonormal Basis

All vectors have length 1 Normalize after find orthogonal basis

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

QR Factorization - Example

 

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

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

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- 9 © 2012 Pearson Education, Inc.

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- 10 © 2012 Pearson Education, Inc.

Length, Distance, Orthogonality

:

||v|| =

 

Example

||p(t)|| and ||q(t)|| from previous example

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

Inequalities

Cauchy Schwartz Inequality:

|<u,v>| ≥ ||u|| ||v||

Triangle Inequality

||u+v|| ≤ ||u|| + ||v||