Review_L1234

# Review_L1234 - MA2213 Review Lectures 1-4 Inner Products...

This preview shows pages 1–5. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Transpose and its Properties Transpose - = - π 2 3 7 4 7 2 3 4 8 3 8 3 T T T T M N MN = ) ( T T M M M ) ( ) ( 0 det 1 1 - - = and Proofs M M T det det = M M T is positive definite 0 ) ( ) ( ) ( 0 = Mv Mv v M M v v T T T Theorem 1 I I MM M M T T T T = = = - - ) ( ) ( 1 1
Inner ( = Scalar) Product Spaces is a vector space R V v u R v u , , ) , ( symmetry V over reals with an inner product that satisfies the following 3 properties: ) , ( ) , ( u v v u = linearity V w v u R b a w u b v u a bw av u + = + , , , , ), , ( ) , ( ) , ( positivity 0 ) , ( 0 u u u Remark Symmetry and Linearity imply ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( u w b u v a w u b v u a bw av u u bw av + = + = + = + hence (- , -) : V x V R is Bilinear

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Examples of Inner Product Spaces Example 2. ( is called a weight function) and ) , 0 ( ) , ( : ]), , ([ = b a b a C V ϖ Example 1. n n R P × positive definite, symmetric Remark The standard inner product on V v u Pv u v u R V T n = , , ) , ( , n R V = is obtained by choosing I P = then j n j j T T v u v u Iv u
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/06/2012 for the course MA 2213 taught by Professor Michael during the Fall '07 term at National University of Singapore.

### Page1 / 10

Review_L1234 - MA2213 Review Lectures 1-4 Inner Products...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online