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
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
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