Why is Matrix Multiplication Not Defined Like This? I'm sure everyone already thought about this at least one time.
Why matrix multiplication is not defined the way showed below?
$$\left( \begin{array}{ccc}
a_{11} & a_{12} & \ldots \\
a_{21} & a_{22} & \ldots \\
\vdots & \vdots & \ddots
\end{array} \right) \cdot
\left( \begin{array}{ccc}
b_{11} & b_{12} & \ldots \\
b_{21} & b_{22} & \ldots \\
\vdots & \vdots & \ddots
\end{array} \right) =
\left( \begin{array}{ccc}
a_{11}\cdot b_{11} & a_{12}\cdot b_{12} & \ldots \\
a_{21}\cdot b_{21} & a_{22}\cdot b_{22} & \ldots \\
\vdots & \vdots & \ddots
\end{array} \right) $$
I know this definition has it's limitations. The product only works with same order matrix and any matrix with some zero entry won't be invertible. But this definition is associative, commutative, has identity element, the distributive works, is simpler and more intuitive.
I have no problem about the classic definition, but this definition also has good properties, why it is never used?
 A: The matrix multiplication we use is defined that way because it corresponds to the composition of linear maps. Recall that, given a vector space $V$ over $K$ with basis $(e_1,\ldots,e_n)$, and a vector space $W$ over $K$ with basis $(f_1,\ldots,f_m)$, we have a natural isomorphism $\eta:Hom_K(V,W) \rightarrow M_{m\times n}(K)$. The map $\eta$ simply sends a linear map to its matrix representation in terms of the two given bases.
This map is more than an isomorphism of vector spaces: it also preserves the algebra structure, in the sense that composition of linear maps is sent to multiplication of the corresponding matrices.
For example, if you were to compute the effect of the composite operations $\mathbb{R}^3 \xrightarrow{p} \mathbb{R}^2 \xrightarrow{r} \mathbb{R}^2$ in terms of their respective matrices, where $p$ is a projection and $r$ a rotation, you'd simply have to multiply the two matrices together.
A: Suppose we used your proposed product, along with the usual addition and subtraction of matrices.  Then all the algebra of $m\times n$ matrices would be the same as if we just used vectors of length $mn$.  (In more detail, you can convert any matrix to a vector by just writing the rows of the matrix, one after the other, as a single long row.  And this rewriting process would preserve all the algebraic structure, namely addition, subtraction, and your multiplication.)  So this algebra of matrices would not really use the 2-dimensional array structure of matrices at all; it would just be a curious ay of writing component-by-component operations on vectors.
