assign7_soln

# assign7_soln - Math 136 Assignment 7 Solutions 1 Show the...

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: Math 136 Assignment 7 Solutions 1. Show the each of the following sets form a basis for the subspace that they span, and determine the coordinates of ~u and ~v with respect to the basis. a) B = 1 1 2 1 , 2 1 1 1 ; ~u = 5 1- 2 1 , ~v = 0 1 3 1 . Solution: Since neither vector in B is a scalar multiple of the other we get that B is linearly independent. Hence, it forms a basis for the subspace which it spans. The coordinates of ~u with respect to B and the coordinates of ~v with respect to B are determined by row-reducing the augmented systems c 1 1 1 2 1 + c 2 2 1 1 1 = 5 1- 2 1 , d 1 1 1 2 1 + d 2 2 1 1 1 = 0 1 3 1 . We make one doubly augmented matrix and row-reduce to get 1 2 5 1 1 1 1 2 1- 2 3 1 1 1 1 ∼ 1 0- 3 2 0 1 4- 1 0 0 0 0 Thus [ ~u ] B = c 1 c 2 =- 3 4 and [ ~v ] B = d 1 d 2 = 2- 1 . b) C = { x 3 + x 2- 1 , x 3 + x + 1 , x 2 + x + 2 } ; ~u = x 3- 3 x 2 + 2 x + 3, ~v =- x 3 + 3 x + 7. Solution: Consider x 3 + 0 x 2 + 0 x + 0 = c 1 ( x 3 + x 2- 1) + c 2 ( x 3 + x + 1) + c 3 ( x 2 + x + 2) 0 = ( c 1 + c 2 ) x 3 + ( c 1 + c 3 ) x 2 + ( c 2 + c 3 ) x + (- c 1 + c 2 + 2 c 3 ) . We row reduce the coefficient matrix of the corresponding system to get 1 1 0 1 0 1 1 1- 1 1 2 ∼ 1 0 0 0 1 0 0 0 1 0 0 0 . Thus, the only solution is c 1 = c 2 = c 3 = 0 so C is linearly independent and hence forms a basis for the subspace which it spans. The coordinates of ~u with respect to B and the coordinates of ~v with respect to B are determined by row-reducing the augmented systems c 1 ( x 3 + x 2- 1) + c 2 ( x 3 + x + 1) + c 3 ( x 2 + x + 2) = x 3- 3 x 2 + 2 x + 3 c 1 ( x 3 + x 2- 1) + c 2 ( x 3 + x + 1) + c 3 ( x 2 + x + 2) =- x 3 + 3 x...
View Full Document

## This note was uploaded on 09/19/2011 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

### Page1 / 5

assign7_soln - Math 136 Assignment 7 Solutions 1 Show the...

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

View Full Document
Ask a homework question - tutors are online