Section 4.7 Ordered basis and linear maps
Subsection 4.7.1 Ordered basis
Remark 4.7.2.
If a sequence \(v_1,v_2,\ldots,v_n\) is an ordered basis for \(V\) then we write it as \((v_1,v_2,\ldots,v_n)\text{.}\) Thus, we consider ordered basis \((v_1,v_2,\ldots,v_n)\) and ordered basis \(\big(v_{\sigma(1)},v_{\sigma(2)},\ldots,v_{\sigma(n)}\big)\) to be distinct for every non-identity permutation \(\sigma\text{.}\)
To put it in different words, in an ordered basis the order in which basis vectors occur is important. For instance, the ordered basis \((v,w)\) is different from \((w,v)\text{.}\) However, as sets \(\{v,w\}\) and \(\{w,v\}\) are the same.
Definition 4.7.3.
Let \((v_1,v_2,\ldots,v_n)\) be an ordered basis of \(V\text{.}\) Given \(v\in V\) there is a unique \(n\)-tuple \((\alpha_1,\alpha_2,\ldots,\alpha_n)\) of scalars such that \(v=\sum_i\alpha_iv_i\) (see Exercise 3.5.11). We call \(\alpha_i\) the \(i\)-th coordinate of \(v\) relative to the ordered basis \((v_1,v_2,\ldots,v_n)\text{.}\)Subsection 4.7.2 Matrix representation of a linear transformation
Let \(V\) and \(W\) be finite-dimensional vector spaces over a field \(F\text{.}\) Suppose that \(\mathfrak{B}_V=(v_1,v_2,\ldots,v_n)\) and \(\mathfrak{B}_W=(w_1,w_2,\ldots,w_m)\) are ordered bases of \(V\) and \(W\text{,}\) respectively. Let \(T\colon V\to W\) be an \(F\)-linear transformation.
Since \(\mathfrak{B}_V\) is a basis, for a given \(v\in V\) there are uniquely determined \(\alpha_i\in F\) such that \(v=\sum\alpha_iv_i\) (see Exercise 3.5.11). Then,
\begin{equation*}
T(v)=T\big(\sum\alpha_iv_i\big)=\sum\alpha_iT(v_i).
\end{equation*}
Hence, \(T\) is completely determined by \(T(v_i)\) for \(1\leq i\leq n\text{.}\)
Because the ordered set \(\mathfrak{B}_W\) is a basis of \(W\) we get, for every \(i\in\{1,2,\ldots,n\}\text{,}\)
\begin{equation*}
T(v_i)=\sum_{k=1}^{m}\beta_{ki}w_k.
\end{equation*}
Therefore,
\begin{equation*}
T(v)=\sum_{i=1}^{n}\alpha_i\sum_{k=1}^{m}\beta_{ki}w_k=\sum_{k=1}^{m}\big(\sum_{i=1}^{n}\alpha_i\beta_{ki}\big)w_k.
\end{equation*}
Therefore, the \(k\)-th coordinate of \(T(v)\) with respect to \(\mathfrak{B}_W\) is
\begin{equation}
\sum_{i=1}^{n}\alpha_i\beta_{ki}\text{.}\tag{4.7.1}
\end{equation}
We obtain the matrix of \(T\) relative to ordered bases \(\mathfrak{B}_V\) and \(\mathfrak{B}_W\text{,}\)
\begin{equation*}
[T]_{\mathfrak{B}_V}^{\mathfrak{B}_W}=\begin{pmatrix}\beta_{11}\amp\beta_{12}\amp\cdots\amp\beta_{1n}\\\beta_{21}\amp\beta_{22}\amp\cdots\amp\beta_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\\beta_{m1}\amp\beta_{m2}\amp\cdots\amp\beta_{mn}\end{pmatrix}.
\end{equation*}
We observe the following.
\begin{equation}
\begin{pmatrix}\beta_{11}\amp\beta_{12}\amp\cdots\amp\beta_{1n}\\\beta_{21}\amp\beta_{22}\amp\cdots\amp\beta_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\\beta_{m1}\amp\beta_{m2}\amp\cdots\amp\beta_{mn}\end{pmatrix}\begin{pmatrix}\alpha_1\\\alpha_2\\\vdots\\\alpha_n\end{pmatrix}=\begin{pmatrix}\sum_{i=1}^{n}\alpha_i\beta_{1i}\\\sum_{i=1}^{n}\alpha_i\beta_{2i}\\\vdots\\\sum_{i=1}^{n}\alpha_i\beta_{mi}\end{pmatrix}\tag{4.7.2}
\end{equation}
In particular, if \(e_k=(\delta_{k1},\delta_{k2},\ldots,\delta_{kn})\) then
\begin{equation}
\begin{pmatrix}\beta_{11}\amp\beta_{12}\amp\cdots\amp\beta_{1n}\\\beta_{21}\amp\beta_{22}\amp\cdots\amp\beta_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\\beta_{m1}\amp\beta_{m2}\amp\cdots\amp\beta_{mn}\end{pmatrix}\cdot e_k^t=\begin{pmatrix}\beta_{1k}\\\beta_{2k}\\\vdots\\\beta_{mk}\end{pmatrix}\tag{4.7.3}
\end{equation}
Thus, the action of \([T]_{\mathfrak{B}_V}^{\mathfrak{B}_W}\) on \(e_k^t\) gives the \(k\)-th column of the matrix.
