## Elementary Linear Algebra 7th Edition

Published by Cengage Learning

# Chapter 4 - Vector Spaces - 4.3 Subspaces of Vector Spaces - 4.3 Exercises - Page 167: 4

#### Answer

$W$ is a vector subspace of $M_{3,2}$.

#### Work Step by Step

Let $u=\left[\begin{array}{ll}{a} & {b} \\ {a+b} & {0}\\ {0}&{c}\end{array}\right],v=\left[\begin{array}{ll}{d} & {e} \\ {d+e} & {0}\\ {0}&{f}\end{array}\right],\in W$ and $c'\in R$ where $W$ the set of all $3\times 2$ of the form $\left[\begin{array}{ll}{a} & {b} \\ {a+b} & {0}\\ {0}&{c}\end{array}\right]$. Now, \begin{align*} u+v&= \left[\begin{array}{ll}{a} & {b} \\ {a+b} & {0}\\ {0}&{c}\end{array}\right]+\left[\begin{array}{ll}{d} & {e} \\ {d+e} & {0}\\ {0}&{f}\end{array}\right]\\ &=\left[\begin{array}{ll}{a+d} & {b+e} \\ {a+b+d+e} & {0}\\ {0}&{c+f}\end{array}\right] \end{align*} which means that $u+v\in W$ and also $c'u= \left[\begin{array}{ll}{c'a} & {c'b} \\ {c'(a+b)} & {0}\\ {0}&{c'c}\end{array}\right]\in W$. Hence, $W$ is a vector subspace of $M_{3,2}$.

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