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Every finite-dimensional linear map can be represented by a matrix. But what about the opposite: Does every matrix correspond to a linear map?

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    $\begingroup$ I'm a little confused how you understand finding a matrix for a linear map (which in turn requires you to think of a matrix as a linear map), but not see how a matrix can be thought of as a linear map. Your question is usually the easy direction of this problem, which is what I find odd. $\endgroup$
    – breeden
    Commented Feb 12, 2014 at 16:03
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    $\begingroup$ The fact that every linear map can be represented by a matrix was infused over and over again as an important fact in my math education. The opposite seems intuitive, but was never specifically touched $\endgroup$
    – Basti
    Commented Feb 12, 2014 at 16:32
  • $\begingroup$ Related. $\endgroup$ Commented Sep 18, 2015 at 3:16

3 Answers 3

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Well, not every matrix, necessarily. We could take matrices of arbitrary sets, for example, which needn't have any associated operations.

However, assuming that you're taking matrices of elements of some field $\Bbb F,$ then the answer is yes. Given any such matrix, say $A,$ if $A$ is $m\times n,$ then the map $T:\Bbb F^n\to\Bbb F^m$ given by $T(\vec x)=A\vec x$ is linear.

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    $\begingroup$ "A matrix is a rectangular array of numbers or other mathematical objects, for which operations such as addition and multiplication are defined." - Wikipedia. I'm hard pressed to call any rectangular array of elements a matrix. $\endgroup$
    – Greebo
    Commented Feb 12, 2014 at 16:03
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    $\begingroup$ That's certainly one definition, and allays the problem nicely. Alternately, one can think of an $m\times n$ matrix as a function on $\{1,...,m\}\times\{1,...,n\}.$ It isn't as nice, to be sure, but it still "looks like a matrix." $\endgroup$ Commented Feb 12, 2014 at 16:06
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    $\begingroup$ Sure, but I'd disagree that "looks like a matrix" and "is a matrix" is the same thing. $\endgroup$
    – Greebo
    Commented Feb 12, 2014 at 16:07
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    $\begingroup$ Feel free! It all depends on if we want to be able to do anything with our matrices, aside from giving a more compact visual form for certain sorts of functions. /shrug/ $\endgroup$ Commented Feb 12, 2014 at 16:10
  • $\begingroup$ Please mention the canonical basis here. $\endgroup$ Commented Jan 3, 2022 at 17:13
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Yes. If you have a $m\times n$ matrix $M$, then this can be seen as a map from $\mathbb{R}^n$ to $\mathbb{R}^m$ by $M(x) = Mx$.

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If $A\in\mathcal M_{n,p}(\Bbb R)$ then the map $$f\colon \Bbb R^p\rightarrow \Bbb R^n,\quad x\mapsto A x$$ is a linear transformation which's represented by the matrix $A$ in the canonical basis.

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  • $\begingroup$ Very well done! $\endgroup$
    – amWhy
    Commented Feb 13, 2014 at 13:41

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