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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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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. – breeden Feb 12 '14 at 16:03
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 – Basti Feb 12 '14 at 16:32
Related. – Cameron Buie Sep 18 '15 at 3:16
up vote 7 down vote accepted

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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"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. – Greebo Feb 12 '14 at 16:03
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." – Cameron Buie Feb 12 '14 at 16:06
Sure, but I'd disagree that "looks like a matrix" and "is a matrix" is the same thing. – Greebo Feb 12 '14 at 16:07
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/ – Cameron Buie Feb 12 '14 at 16:10

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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Very well done! – amWhy Feb 13 '14 at 13:41

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