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: 41


$S$ is a basis for $P_3$. .

Work Step by Step

The set $S=\left\{t^{3}-2 t^{2}+1, t^{2}-4, t^{3}+2 t, 5 t\right\}$ is a basis for $P_3$. Indeed, consider the following linear combination $$a(t^{3}-2 t^{2}+1)+b( t^{2}-4)+c(t^{3}+2 t)+d ( 5 t)=0.$$ Comparing the coefficients, we have \begin{align*} a-4b&=0\\ 2c+5d&=0\\ -2a+b&=0\\ a+c&=0. \end{align*} The augmented matrix of the above system is given by $$\left[\begin{array}{cccc} {1}&{-4}&{0}&{0}\\ {0}&{0}&{2}&{5}\\ {-2}&{1}&{0}&{0}\\ {1}&{0}&{1}&{0}\\ \end{array}\right]$$ and its determinant is -35. Therefore, the above system has unique solution, that is, the trivial solution. Then $S$ is linearly independent set of vectors and since $P_3$ has dimension $4$, then by Theorem 4.12, $S$ is a basis for $P_3$.
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