lect136_11_w09

# lect136_11_w09 - Wednesday, January 28 - Lecture 11 :...

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

Wednesday, January 28 Lecture 11 : Subspaces II . (Refers to section 3.4) Expectations : 1. Determine if a given subset of R n is a subspace. 2. Given a subspace W of R n define the orthogonal complement W of W and show it is a subspace. 3. Show that for W = span{ v 1 , v 2 , . ., v m } in R n , x W iff < v i , x > = 0 for all i = 1 to m . 11.1 Theorem If U and W are two subspaces of R n then U W, (the set of all the elements they have in common), is subspace of R n . Proof : o The set U W is non-empty. (Why?) o Let v and w be any two vectors in U W , and let α and β be a scalars. (The two vectors v and w may be the same vector.) o Then belongs to both U and W . o Since both U and W are subspaces then α v + β w belongs to both U and W . So α v + β w belongs to U W and so is a subspace of R n . 11.2 Exercise – Is the set S = {( a , b , – b , c ) : a , b , c in R } a subspace of R 4 ? Find a spanning family for S (if there is one). Solution outline:

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 note was uploaded on 09/04/2009 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

### Page1 / 3

lect136_11_w09 - Wednesday, January 28 - Lecture 11 :...

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

View Full Document
Ask a homework question - tutors are online