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: 19


The set $S=\{(0,0,0),(1,0,0),(0,1,0)\}$ is not a basis for $R^3$.

Work Step by Step

The set $S=\{(0,0,0),(1,0,0),(0,1,0)\}$ is not a basis for $R^3$ because the vectors in $S$ are not linearly independent. Indeed, one can see that the following linear combination $$a(0,0,0)+b(1,0,0)+c(0,1,0)=(0,0,0), \quad a,b,c\in R$$ is valid for any non-zero value of $a$ and $b=c=0$, hence the vectors of $S$ are not linearly independent. Hence, the set $S=\{(0,0,0),(1,0,0),(0,1,0)\}$ is not a basis for $R^3$.
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