Lecturenotes16-18March2011

# Lecturenotes16-18March2011 - Relation of row space and...

This preview shows pages 1–2. Sign up to view the full content.

Relation of row space and column space of A, to Ax=b Theorem 4.7.1: A system of linear equations Ax=b is consistent if and only if b is in the column space of A. Idea: Consider Ax = b where x = x 1 . . . x n , A = ± c 1 ... c n ² . c 1 ,...,c n denotes the columns of A. Then we can write Ax = x 1 c 1 + x 2 c 2 + ... + x n c n = b so if Ax=b is consistent this means there are x 1 ,x 2 ,...,x n satisfying the above equation. This means b is in the column space of A. On the other hand if b is in the column space of A then it means we can write b = c 1 x 1 + ... + c n x n which means Ax=b has a solution. Theorem 4.8.5: If Ax=b is a consistent linear system of m equations in n unknowns, and if A has rank r, then the general solution of the system contains n-r parameters. Idea: rank(A)=r=number of leading variables, hence there will be n-r free variables. Deﬁnition: If W is a subspace of R n , then the set of all vectors in R n that are orthog- onal to every vector in W is called the orthogonal complement of W and is denoted by W . Example: (a) Consider the plane x + 2 y + 3 z = 0 . This plane is a vector space. Its orthogonal complement is the line represented by the vector equation x = t (1 , 2 , 3) where t R . (b) L1: x=t(1,2) and L2: t(-2,1) are vector spaces orthogonal complement of each other. Example: Find a basis and dimension of W where W is a plane in R 3 passing through origin and parallel to the vectors v 1 = (1 , 2 , 3) and v 2 = (4 , 0 , 2) . Solution: W = { x = t 1 v 1 + t 2 v 2 | t 1 ,t 2 R } , then W = { v R 3 | v · x = 0 , for all x W } . 1st way: We want

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This document was uploaded on 04/26/2011.

### Page1 / 4

Lecturenotes16-18March2011 - Relation of row space and...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online