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.