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 - 4.5 Basis and Dimension - 4.5 Exercises - Page 187: 34


$S$ is a basis for $R^2$.

Work Step by Step

The set $S=\{(1,2),(1,-1)\}$ is a linearly independent set of vectors. Indeed, assume that $$a(1,2)+b(1,-1)=(0,0), \quad a,b\in R.$$ Then, we have the following system of equations \begin{align*} a+b&=0\\ 2a-b&=0. \end{align*} Solving the above equations, we find that $a=0,b=0$ and hence $S$ is linearly independent. Now, since $R^2$ is vector space of dimension $2$, then by Theorem 4.12, $S$ is a basis for $R^2$.
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