Question

(a) Let $A$ be an $m \times n$ matrix and let $V$ be a subspace of $\mathbb{R}^n$. Show that $W=A V=$ $\{A \mathbf{v} \mid \mathbf{v} \in V\}$ forms a subspace of $\operatorname{img} A$. (b) If $\operatorname{dim} V=k$, show that $\operatorname{dim} W \leq \min \{k, r\}$, where $r=\operatorname{rank} A$. Give an example in which $\operatorname{dim}(A V)<\operatorname{dim} V$.

    (a) Let $A$ be an $m \times n$ matrix and let $V$ be a subspace of $\mathbb{R}^n$. Show that $W=A V=$ $\{A \mathbf{v} \mid \mathbf{v} \in V\}$ forms a subspace of $\operatorname{img} A$. (b) If $\operatorname{dim} V=k$, show that $\operatorname{dim} W \leq \min \{k, r\}$, where $r=\operatorname{rank} A$. Give an example in which $\operatorname{dim}(A V)<\operatorname{dim} V$.

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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 2, Problem 46

Instant Answer

Step 1

- To show that $W$ is a subspace, we need to verify that it is closed under addition and scalar multiplication, and that it contains the zero vector. - **Closure under addition**: Take any two vectors $u, v \in V$. Since $V$ is a subspace, $u + v \in V$. Show more…

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(a) Let $A$ be an $m \times n$ matrix and let $V$ be a subspace of $\mathbb{R}^n$. Show that $W=A V=$ $\{A \mathbf{v} \mid \mathbf{v} \in V\}$ forms a subspace of $\operatorname{img} A$. (b) If $\operatorname{dim} V=k$, show that $\operatorname{dim} W \leq \min \{k, r\}$, where $r=\operatorname{rank} A$. Give an example in which $\operatorname{dim}(A V)<\operatorname{dim} V$.

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