1. Gram-Schmidt Orthogonalization
MA2213 Review Lectures 1-4
Inner Products
Gramm Matrices
Gram-Schmidt Orthogonalization

2. Transpose and its Properties
Theorem 1
and
is positive definite
Proofs

3. Inner ( = Scalar) Product Spaces
is a vector space
over reals
with an inner product
that satisfies the following 3 properties:
symmetry
linearity
positivity
Remark Symmetry and Linearity imply
hence (- , -) : V x V R is Bilinear

4. Examples of Inner Product Spaces
positive definite, symmetric
Remark The standard inner product on
is obtained by choosing
then
Example 2.
( is called a weight function) and
Remark The SIP on
is obtained by choosing

5. The Gramm Matrix
of a finite sequence of vectors
in an inner product space V is the
matrix
Theorem 2 Let
are the columns vectors of
Then
is the Gramm matrix of the
sequence
Proof

6. Standard Basis
Definition The standard (sequence)
of basis vectors for is
where

7. Questions
Question 1. What is the following matrix
Question 2. What is the following ? if
Question 2. For the standard inner
product on
what is ?

8. Gram-Schmidt Orthogonalization
Theorem 3. Given a sequence of linearly independent vectors in an inner
product space
there exists a unique upper
triangular matrix
with diagonal entries 1
such that the ‘matrix’
has orthogonal column vectors.
Proof Since
it suffices to show that
for
these are n-1 systems with Gramm matrices

9. Gram-Schmidt Algorithm
start with

10. Gram-Schmidt Orthonormalization
produce an orthonormal
basis
start with
Here