## Elementary Linear Algebra 7th Edition

Published by Cengage Learning

# Chapter 4 - Vector Spaces - 4.6 Rank of a Matrix and Systems of Linear Equations - 4.6 Exercises - Page 199: 15

#### Answer

A basis for the subspace is given by $$\{2,9,-2,53),(0,31,0,155),(0,0,0,1)\}.$$

#### Work Step by Step

Let $S$ be given by $$S=\{ (2,9,-2,53),(-3,2,3,-2),(8,-3,-8,17)(0,-3,0,15) \}.$$ We form the matrix $$\left[ \begin {array}{ccccc} 2&9&-2&53\\ -3&2&3&-2\\ 8&-3&-8&17\\0&-3&0&15 \end {array} \right].$$ The reduced form of the matrix is given by $$\left[ \begin {array}{cccc} 2&9&-2&53\\ 0&31&0&155\\ 0&0&0&30 \\ 0&0&0&0\end {array} \right] .$$ A basis for the subspace is given by $$\{2,9,-2,53),(0,31,0,155),(0,0,0,1)\}.$$

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