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: 23


$S$ does not span $P_2$.

Work Step by Step

The set $S=\left\{1-x, 1-x^{2}, 3 x^{2}-2 x-1\right\}$ is not a basis for $P_2$ because $S$ does not span $P_2$. For example, consider the constant function $f(x)=1$ as a linear combination of the elements of $S$ as follows $$1=a(1-x)+b(1-x^{2})+c(3 x^{2}-2 x-1).$$ Comparing the coefficients, we have \begin{align*} a+b-c&=1\\ -a-2c&=0\\ -b+3c&=0. \end{align*} It is easy to note that the above system is in-consistent, since we have $a=-2c$ and $b=3c$. Now, substituting into the first equation we get $0=1$ which is a contradiction and hence the system is in-consistent.
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