Let $m$ and $n$ be natural numbers.

We have not only seen that an $(m\times n)$-matrix $A$ determines a linear map $L_A:\mathbb{R}^n\to \mathbb{R}^m$, but also that each linear map $L:V\to W$ from a vector space $V$ with basis $\alpha$ of length $n$ to a vector space $W$ with basis $\beta$ of length $m$ can be written as the composition $L = \beta^{-1} L_A\alpha$ of three linear maps and as $L = L_A$ where $A$ is the $(m\times n)$-matrix $\left(\beta L\alpha^{-1}\right)_{\varepsilon}$. Here, $\beta$ and $\alpha$ are the coordinatizations with respect to the bases of the same name. The matrix $A$ describes the linear map corresponding to $L$ between the coordinate spaces. Thus, the matrix $A$ completely determines the linear mapping $L$.

We have also seen that the operations on linear transformations can be expressed well in terms of the corresponding matrices. We summarize this in the following table, where, for $(m\times n)$-matrices $A$ and $B$, the associated linear maps are written as follows: $L = \beta^{-1} L_A\alpha$ and $M = \beta^{-1} L_B\alpha$.

$$\begin{array}{lc|r|cl}\text{operation}&&\text{matrix}&&\text{linear map }&&\\ \hline \text{sum}&&A+B&&L+M\\ \text{scalar multiple}&&\lambda A&&\lambda\cdot L\\ \text{composition}&&A\,B&&L\,M\\ \text{inverse}&&A^{-1}&&L^{-1}\\ \text{transposed}&&A^{\top}&&L^\star\\ \hline\end{array}$$

The latter is new and will be treated later, in the context of dual spaces.