Linear Algebra: What matrices are
I actually object a little bit to the claim that matrices <i>are</i> representations of linear transformations. No, matrices are just two-dimensional arrays of numbers. If you then define a specific 'mutliplication' operation on those arrays, and a mapping from linear functions to matrices, it turns out that that multiplication operation is isomorphic to function application. That's neat! But it doesn't mean that that is what matrices <i>are</i>.<p>If you came up with an isomorphism from matrices to a domain where it made sense to define a different multiplication operation - like maybe placewise multiplication, where<p><pre><code>   [a b] * [e f] = [ae bf]
   [c d]   [g h]   [cg dh]
</code></pre>
then that would be just as valid, but it wouldn't change what matrices 'are'. In fact, because you know how to map matrices to linear functions, it would let you describe an operation to combine two linear functions in a new way and that might lead to some new insight about linear algebra!<p>It's like how, in school you were taught that you can't multiply vectors together. Yet, in shader languages, it turns out that it's <i>really useful</i> to be able to multiply two vectors just by multiplying each component ([a,b][c,d]=[ab,cd]), so they define that as a valid operation.