The document discusses orthogonal bases and the Gram-Schmidt process. It defines the Gram-Schmidt process as an algorithm for finding an orthogonal basis from a given basis by making each new vector orthogonal to the previous ones. It also discusses orthonormal bases, QR factorization, inner products, length and distance, and applying Gram-Schmidt to produce an orthogonal basis for the vector space of polynomials up to degree 2.

1. Orthogonal Basis
.
Find an orthogonal basis for W.


2. 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

3. Gram-Schmidt - Example
.
Find an orthogonal basis for W.


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

5. 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

6. QR Factorization - Example


7. 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

8. 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

9. 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.

10. 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.

11. Length, Distance, Orthogonality
:
||v|| =


12. Example
 ||p(t)|| and ||q(t)|| from previous example

13. 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

14. Inequalities
 Cauchy Schwartz Inequality:
|<u,v>| ≥ ||u|| ||v||
 Triangle Inequality
||u+v|| ≤ ||u|| + ||v||