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
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 ~x and ~ y with respect to the basis. a) { (1 , 1 , , 1 , 0) , (1 , , 2 , 1 , 1) , (0 , , 1 , 1 , 3) } ; ~x = (2 , 2 , 5 , 1 , 5), ~ y = ( 1 , 3 , 3 , 2 , 1). Consider 0 = c 1 (1 , 1 , , 1 , 0) + c 2 (1 , , 2 , 1 , 1) + c 3 ((0 , , 1 , 1 , 3) = ( c 1 + c 2 ,c 1 , 2 c 2 + c 3 ,c 1 + c 2 + c 3 ,c 2 + 3 c 3 ) We rowreduce the coefficient matrix to get 1 1 0 1 0 0 0 2 1 1 1 1 0 1 3 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 . Thus, the only solution is c 1 = c 2 = c 3 = 0 so the set of vectors is linearly independent and so the vectors in B form a basis for the subspace which they span. The coordinates of ~x with respect to B and the coordinates of ~ y with respect to B are determined by rowreducing the augmented systems c 1 (1 , 1 , , 1 , 0) + c 2 (1 , , 2 , 1 , 1) + c 3 (0 , , 1 , 1 , 3) = (2 , 2 , 5 , 1 , 5) d 1 (1 , 1 , , 1 , 0) + d 2 (1 , , 2 , 1 , 1) + d 3 (0 , , 1 , 1 , 3) = ( 1 , 3 , 3 , 2 , 1) . We make one doubly augmented matrix and rowreduce to get 1 1 0 2 1 1 0 0 2 3 0 2 1 5 3 1 1 1 1 2 0 1 3 5 1 1 0 0 2 3 0 1 0 4 2 0 0 1 3 1 0 0 0 0 0 0 Thus [ ~x ] B = c 1 c 2 c 3 =  2 4 3 and [ ~ y ] B = d 1 d 2 d 3 =  3 2 1 . b) 1 1 1 0 , 0 1 1 1 , 2 1 ; ~x = 0 1 1 2 , ~ y = 4 1 1 4 ....
View Full
Document
 Spring '08
 All
 Linear Algebra, Algebra, Sets

Click to edit the document details