Linear Algebra/Any Matrix Represents a Linear Map

The prior subsection shows that the action of a linear map $$h$$ is described by a matrix $$H$$, with respect to appropriate bases, in this way.



\vec{v}=\begin{pmatrix} v_1 \\ \vdots \\ v_n \end{pmatrix}_B \;\overset{h}{\underset{H}{\longmapsto}}\; \begin{pmatrix} h_{1,1}v_1+\dots+h_{1,n}v_n \\ \vdots                     \\ h_{m,1}v_1+\dots+h_{m,n}v_n \end{pmatrix}_D =h(\vec{v}) $$

In this subsection, we will show the converse, that each matrix represents a linear map.

Recall that, in the definition of the matrix representation of a linear map, the number of columns of the matrix is the dimension of the map's domain and the number of rows of the matrix is the dimension of the map's codomain. Thus, for instance, a $$ 2 \! \times \! 3 $$ matrix cannot represent a map from $$ \mathbb{R}^5 $$ to $$ \mathbb{R}^4 $$. The next result says that, beyond this restriction on the dimensions, there are no other limitations: the $$ 2 \! \times \! 3 $$ matrix represents a map from any three-dimensional space to any two-dimensional space.

So not only is any linear map described by a matrix but any matrix describes a linear map. This means that we can, when convenient, handle linear maps entirely as matrices, simply doing the computations, without have to worry that a matrix of interest does not represent a linear map on some pair of spaces of interest. (In practice, when we are working with a matrix but no spaces or bases have been specified, we will often take the domain and codomain to be $$\mathbb{R}^n$$ and $$\mathbb{R}^m$$ and use the standard bases. In this case, because the representation is transparent&mdash; the representation with respect to the standard basis of $$\vec{v}$$ is $$\vec{v}$$&mdash; the column space of the matrix equals the range of the map. Consequently, the column space of $$ H $$ is often denoted by $$ \mathcal{R}(H) $$.)

With the theorem, we have characterized linear maps as those maps that act in this matrix way. Each linear map is described by a matrix and each matrix describes a linear map. We finish this section by illustrating how a matrix can be used to tell things about its maps.

The above results end any confusion caused by our use of the word "rank" to mean apparently different things when applied to matrices and when applied to maps. We can also justify the dual use of "nonsingular". We've defined a matrix to be nonsingular if it is square and is the matrix of coefficients of a linear system with a unique solution, and we've defined a linear map to be nonsingular if it is one-to-one.

We've now seen that the relationship between maps and matrices goes both ways: fixing bases, any linear map is represented by a matrix and any matrix describes a linear map. That is, by fixing spaces and bases we get a correspondence between maps and matrices. In the rest of this chapter we will explore this correspondence. For instance, we've defined for linear maps the operations of addition and scalar multiplication and we shall see what the corresponding matrix operations are. We shall also see the matrix operation that represent the map operation of composition. And, we shall see how to find the matrix that represents a map's inverse.

Exercises
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