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 - 4.4 Spanning Sets and Linear Independence - 4.4 Exercises - Page 178: 6


$$\left[ \begin {array}{cc} 6&2\\ 9&11 \end {array} \right]=3\left[ \begin {array}{cc} 2&-3\\ 4&1 \end {array} \right]-4\left[ \begin {array}{cc} 0&5\\ 1&-2 \end {array} \right].$$

Work Step by Step

Assume the combination $$\left[ \begin {array}{cc} 6&2\\ 9&11 \end {array} \right]=a\left[ \begin {array}{cc} 2&-3\\ 4&1 \end {array} \right]+b\left[ \begin {array}{cc} 0&5\\ 1&-2 \end {array} \right].$$ We have the system \begin{align*} 2a&=6\\ -3a+5b&=2\\ 4a+b&=9\\ a-2b&=11. \end{align*} We get the solution $$a=3, \quad b=-4.$$ Consequently, $$\left[ \begin {array}{cc} 6&2\\ 9&11 \end {array} \right]=3\left[ \begin {array}{cc} 2&-3\\ 4&1 \end {array} \right]-4\left[ \begin {array}{cc} 0&5\\ 1&-2 \end {array} \right].$$
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