# A3S - ASSIGNMENT 3 Solutions For i = 1 ,...,k , let A i be...

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

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.

Unformatted text preview: ASSIGNMENT 3 Solutions For i = 1 ,...,k , let A i be an n i × n i +1 matrix whose rows span R n i +1 . 1. [3 marks] Prove that the rows of A 1 ··· A k span R n k +1 . We have ( A 1 ··· A k ) T = A T k ··· A T 1 . It suffices to show that the columns of the above matrix span R n k +1 . This is equivalent to showing that A T k ··· A T 1 x = b has a solution for any b ∈ R n k +1 . Let b ∈ R n k +1 be given. We apply the following chain of reasoning: Since the columns of A T k span R n k +1 , A T k x = b has a solution y k ∈ R n k . Since the columns of A T k- 1 span R n k , A T k- 1 x = y k has a solution y k- 1 ∈ R n k- 1 . Since the columns of A T k- 2 span R n k- 1 , A T k- 2 x = y k- 1 has a solution y k- 2 ∈ R n k- 2 . . . . Since the columns of A T 2 span R n 3 , A T 2 x = y 3 has a solution y 2 ∈ R n 2 . Since the columns of A T 1 span R n 2 , A T 1 x = y 2 has a solution y 1 ∈ R n 1 ....
View Full Document

## This note was uploaded on 04/07/2010 for the course MATH 136 taught by Professor All during the Fall '08 term at Waterloo.

### Page1 / 3

A3S - ASSIGNMENT 3 Solutions For i = 1 ,...,k , let A i be...

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

View Full Document
Ask a homework question - tutors are online