Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 4 - Vector Spaces - 4.6 Rank of a Matrix and Systems of Linear Equations - 4.6 Exercises - Page 199: 14


A basis for the subspace is given by $$\{1,2,2),(0,0,1)\}.$$

Work Step by Step

Let $S$ be given by $$ S=\{(1,2,2),(-1,0,0),(1,1,1)\} $$ We form the matrix $$ \left[ \begin {array}{ccccc} 1&2&2\\ -1&0&0\\ 1&1&1 \end {array} \right]. $$ The reduced form of the matrix is given by $$\left[ \begin {array}{ccc} 1&1&2\\ 0&1&2 \\ 0&0&1\end {array} \right] .$$ A basis for the subspace is given by $$\{1,2,2),(0,0,1)\}.$$
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