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Section 5.2 Matrices and a space of linear transformations

In this section we give dictionary between the space of all \(m\times n\) matrices over a field \(F\) and the space of all \(F\)-linear transformations between vector spaces \(V\) and \(W\) of dimensions \(n\) and \(m\text{,}\) respectively. We begin with the following observations.

Let \(A\) be a matrix representation of a linear transformation \(S\colon V\to W\) with respect to bases \(\mathfrak{B}_V\) and \(\mathfrak{B}_W\text{.}\) By Proposition 5.1.9, there are \(F\)-isomorphisms

\begin{equation*} T_{\mathfrak{B}_V}\colon V\xrightarrow{\sim} F^n\quad\text{and}\quad T_{\mathfrak{B}_W}\colon W\xrightarrow{\sim} F^m. \end{equation*}

By Checkpoint 5.1.8 we have \(F\)-isomorphisms

\begin{equation*} _nT\colon F^n\xrightarrow{\sim} M_{n\times 1}(F)\quad\text{and}\quad _mT\colon F^m\xrightarrow{\sim} M_{m\times 1}(F). \end{equation*}

Consider the following composition of maps.

\begin{equation} V\xrightarrow{T_{\mathfrak{B}_V}}F^n\xrightarrow{_nT}M_{n\times 1}(F)\xrightarrow{L_A}M_{m\times 1}(F)\xrightarrow{_mT^{-1}}F^{m}\xrightarrow{T_{\mathfrak{B}_W}^{-1}}W\label{composition}\tag{5.1} \end{equation}

The map \(L_A\) is the left multiplication by the matrix \(A\) (see Example 4.2.2). The composition above is denoted by \(\ell_A\text{.}\) Thus,

\begin{equation} \ell_A=T_{\mathfrak{B}_W}^{-1}\circ{} _mT^{-1}\circ L_A\circ{} _nT\circ T_{\mathfrak{B}_V}.\label{left-multi-arbitrary-matrix}\tag{5.2} \end{equation}

By eq. (4.2) it follows that the given linear transformation \(S\) is the same as the composition transformation given in eq. (5.1). Furthermore, given a matrix \(A\in M_{m\times n}(F)\) we can define \(\ell_A\) using Example 4.2.2 and eq. (5.2).

Remark 5.2.1.
Suppose that \(\mathfrak{B}_V=(v_1,\ldots,v_n)\) and \(\mathfrak{B}_W=(w_1,\ldots,w_m)\) are ordered bases for \(V\) and \(W\text{,}\) respectively. The composition \({}_nT\circ T_{\mathfrak{B}_V}\colon V\to M_{n\times 1}(F)\) in the eq. (5.2) is given by
\begin{equation*} \sum\alpha_iv_i\xrightarrow{{}_nT\circ T_{\mathfrak{B}_V}}\begin{pmatrix}\alpha_1\\\alpha_2\\\vdots\\\alpha_n\end{pmatrix}. \end{equation*}
While the composition \(T^{-1}_{\mathfrak{B}_W}\circ {}_mT^{-1}\colon M_{m\times 1}(F)\to W\) in eq. (5.2) is given by
\begin{equation*} \begin{pmatrix}\beta_1\\\beta_2\\\vdots\\\beta_m\end{pmatrix}\xrightarrow{T^{-1}_{\mathfrak{B}_V}\circ {}_mT^{-1}}\sum\beta_iw_i. \end{equation*}
Consider vector spaces \(V=M_2(\R)\) and \(W=\R^3\) over \(\R\text{.}\) Let \(\mathfrak{B}_V=(E_{11},E_{12},E_{21},E_{22})\) and \((e_1,\ldots,e_4)\) be standard bases for \(V\) and \(W\text{,}\) respectively. Consider the following matrix.
\begin{equation*} A=\begin{pmatrix}1\amp 0\amp 0\amp 0\\0\amp 1\amp 0\amp 0\\0\amp 0\amp 0\amp 0\end{pmatrix}\in M_{3\times 4}(\R) \end{equation*}
The map
\begin{equation*} {}_4T\circ T_{\mathfrak{B}_V}\colon V\to M_{4\times 1}(\R) \end{equation*}
is given by
\begin{equation*} E_{11}\mapsto\begin{pmatrix}1\\0\\0\\0\end{pmatrix},\;E_{12}\mapsto\begin{pmatrix}0\\1\\0\\0\end{pmatrix},\;E_{21}\mapsto\begin{pmatrix}0\\0\\1\\0\end{pmatrix},\;E_{22}\mapsto\begin{pmatrix}0\\0\\0\\1\end{pmatrix}. \end{equation*}
The map
\begin{equation*} L_A\colon M_{4\times 1}(\R)\to M_{3\times 1}(\R) \end{equation*}
is given by
\begin{equation*} A\begin{pmatrix}1\\0\\0\\0\end{pmatrix}=\begin{pmatrix}1\\0\\0\end{pmatrix},\; A\begin{pmatrix}0\\1\\0\\0\end{pmatrix}=\begin{pmatrix}0\\1\\0\end{pmatrix},\; A\begin{pmatrix}0\\0\\1\\0\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix},\;A\begin{pmatrix}0\\0\\0\\1\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix}. \end{equation*}
The map
\begin{equation*} T^{-1}_{\mathfrak{B}_W}\circ{}_3T^{-1}\colon M_{3\times 1}(\R)\to W \end{equation*}
is given by
\begin{equation*} \begin{pmatrix}1\\0\\0\end{pmatrix}\mapsto e_1,\;\begin{pmatrix}0\\1\\0\end{pmatrix}\mapsto e_2,\;\begin{pmatrix}0\\0\\1\end{pmatrix}\mapsto e_3. \end{equation*}
Combining all the above maps we obtained that the \(\ell_A\colon M_2(\R)\to\R^3\) is given by
\begin{equation*} E_{11}\mapsto e_1,\; E_{12}\mapsto e_2,\; E_{21}\mapsto 0,\; E_{22}\mapsto 0. \end{equation*}

Fixing bases \(\mathfrak{B}_V\) and \(\mathfrak{B}_W\) we have defined following maps: one is

\begin{gather*} \big\{F\text{-linear transformations }V\to W\big\}\longrightarrow M_{m\times n}(F)\\ T\mapsto [T]_{\mathfrak{B}_V}^{\mathfrak{B}_W} \end{gather*}

and the other one is

\begin{gather*} M_{m\times n}(F)\longrightarrow\big\{F\text{-linear transformations }V\to W\big\}\\ A\mapsto \ell_A \end{gather*}

The next theorem shows that these maps are in fact \(F\)-linear inverses of each other.

It is left to the reader to check that \(\ell\) is an \(F\)-linear transformation. The observations made above eq. (5.2) shows that \(\ell\) is surjective. We show the injectivity of \(\ell\) by showing \(\ker(\ell)=\{0\}\) (see Lemma 4.4.4). Suppose that \(A\in M_{m\times n}(F)\) is such that \(\ell_A=0\text{.}\) Therefore, for any \(v\in V\) we have
\begin{equation*} \big(T_{\mathfrak{B}_W}^{-1}\circ{} _mT^{-1}\circ L_A\circ{} _nT\circ T_{\mathfrak{B}_V}\big)(v)=0\quad\text{i.e.,}\quad T_{\mathfrak{B}_W}^{-1}\circ{} _mT^{-1}\circ L_A({} _nT\circ T_{\mathfrak{B}_V}(v))=0. \end{equation*}
Since, \(T_{\mathfrak{B}_W}^{-1}\) and \({}_mT^{-1}\) are isomorphisms, we get \(L_A({} _nT\circ T_{\mathfrak{B}_V}(v))=0\) for every \(v\in V\text{.}\) Furthermore, \({} _nT\) and \(T_{\mathfrak{B}_V}\) are isomorphisms, and hence their composition is surjective. Therefore, \(L_A=0\text{.}\) Recall that \(\{e_i^t=(\delta_{i1},\delta_{i2},\ldots,\delta_{in})^t\}\) is a basis for \(M_{n\times 1}(F)\text{,}\) and \(L_A(e_i^t)=0\) is the \(i\)-th column of \(A\) (see eq. (4.3)). Hence, \(A=0\text{,}\) and \(\ell\) is injective.