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: 24


$S$ spans $R^3$.

Work Step by Step

Assume the combination $$a(1,0,1)+b(1,1,0)+c(0,1,1)=(0,0,0).$$ We have the system \begin{align*} a+b&=0\\ b+c&=0\\ a+c&=0 \end{align*} Since the determinant of the matrix is given by $$\left| \begin{array} {cc} 1&1&0\\0&1&1\\1&0&1 \end{array} \right|=2 \neq 0$$ then the system has unique solution, that is, $$a=0, \quad b=0, \quad c=0.$$ Consequently, $S$ is linearly independent and since $R^3$ has the dimension $3$ then $S$ spans $R^3$.
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