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Let $T$ : $\mathbb{R}^n \rightarrow \mathbb{R}^m$ be a matrix transformation given by

$T(x) = Ax$ ,

where $A$ is an $m \times n$ real matrix. Assuming $\mathbb{R}^n $ and $\mathbb{R}^m$ are given their standard Euclidean norms, prove that $T$ is continuous.

Not really too sure how to go about proving this, in this situation could you use the fact that if a linear map $T:V \rightarrow W$, with $V$ and $W$ normed vector spaces, is continous at $0$ then $T$ is continous?

Jean Marie
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Brum
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    What you will want to show is $A$ is bounded in the sense that there is some $M>0$ such that $|Ax|\leq M|x|$ for all $x$. Then show that this implies continuity. – Aweygan Aug 14 '16 at 06:15

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