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: 19


$S$ spans $R^2$.

Work Step by Step

Assume the combination of two vectors only of $S$ $$a(-1,4)+b(1,1)=(0,0).$$ We have the system \begin{align*} -a+b&=0\\ 4a+b&=0. \end{align*} Since the determinant of the matrix is given by $$\left| \begin{array} {cc} -1&1\\4&1\end{array} \right|=-5\neq 0$$ then the system has unique solution, that is, $$a=0, \quad b=0.$$ Consequently, the vectors $(-1,4),(1,1)$ are linearly independent and since $R^2$ has the dimension $2$ then they span $R^2$ and hence $S$ spans $R^2$.
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