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: 25


S does not span $R^3.$

Work Step by Step

Assume the combination $$a(1,-2,0)+b(0,0,1)+c(-1,2,0)=(x,,y,z).$$ We have the system \begin{align*} a-c&=x\\ -2a+2c&=y\\ b&=z. \end{align*} The above system has a solution, if the determinant of the coefficient matrix is non zero but we have $$\left| \begin{array} {cc} 1&0&-1\\-2&0&2\\0&1&0 \end{array} \right|= 0.$$ Consequently, the system has no solutions and so $S$ does not span $R^3$.
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