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: 22


$S$ spans $R^3$.

Work Step by Step

Assume the combination $$a(6,7,6)+b(3,2,-4)+c(1,-3,2)=(0,0,0).$$ We have the system \begin{align*} 6a+3b+c&=0\\ 7a+2b-3c&=0\\ 6a-4b+2c&=0 \end{align*} Since the determinant of the matrix is given by $$\left| \begin{array} {cc} 6&3&1\\7&2&-3\\6&-4&2 \end{array} \right|=-184\neq 0$$ 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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