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

Transpose and its Properties Theorem 1 and is positive definite Proofs

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

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

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

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

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 ?

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

Gram-Schmidt Algorithm start with

Gram-Schmidt Orthonormalization produce an orthonormal basis start with Here