Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 4 - Vector Spaces - 4.2 Exercises - Page 208: 14

Answer

By Theorem 3, W is a vector space.

Work Step by Step

$W\subset \mathbb{R}^{3}$. $u\in W\Rightarrow u=a\left[\begin{array}{l} -1\\ 1\\ 3 \end{array}\right]+b\left[\begin{array}{l} 2\\ -2\\ -6 \end{array}\right]$ $W=$Span $\{ \left[\begin{array}{l} -1\\ 1\\ 3 \end{array}\right],\ \left[\begin{array}{l} -1\\ 1\\ 3 \end{array}\right] \}$ By definition, on p.203 The column space of an m$\times$n matrix A, written as Col A, is the set of all linear combinations of the columns of A. $W=$Col A, where A=$\left[\begin{array}{ll} -1 & 2\\ 1 & 1\\ 3 & 3 \end{array}\right]$ By Theorem 3, The column space of an m$\times$n matrix A is a subspace of $\mathbb{R}^{m}$. So W is a vector space.
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