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 - Review Exercises - Page 221: 32


(a) $S$ spans $R^3$. (b) $S$ is not linearly independent set of vectors. (c) $S$ is not a basis for $R^3$.

Work Step by Step

Let $S$ be given by $$ S=\{(1,0,0),(0,1,0),(0,0,1),(2,-1,0)\}.$$ (a) For any $u=(x,y,z\in R^3$, one can write it as follows $$u=(x,y,z)=x(1,0,0)+y(0,1,0)+z(0,0,1)+0(2,-1,0), \quad a,b,c\in R.$$ Then, $S$ spans $R^3$. (b) Since one have the following combination $$(2,-1,0)=2(1,0,0)-(0,1,0)+0(0,0,1)$$ then $S$ is not linearly independent set of vectors. (c) Since $S$ is not linearly independent set, then it is not a basis for $R^3$.
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