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Questions for Math-1: A course for Biology and Applied Geology students

Abhay Soman

Section 1.7 Matrices and linear maps

Definition 1.7.1.

Let \(m,n\in\N\text{.}\) A function \(T\colon M_{n\times 1}(\K)\to M_{m\times 1}(\K)\) is said to be a linear map if it satisfies the following conditions.
  1. For any \(\lambda\in\K\) and any \(v\in M_{n\times 1}(\K)\text{,}\)
    \begin{equation*} T(\lambda\, v)=\lambda\,T(v)\in M_{m\times 1}(\K). \end{equation*}
  2. For any \(v,w\in M_{n\times 1}(\K)\text{,}\)
    \begin{equation*} T(v+w)=T(v)+T(w)\in M_{m\times 1}(\K). \end{equation*}

Definition 1.7.2.

Let \(n\in\N\text{.}\) For each \(i\in\{1,2,3,\ldots,n\}\) we consider the following column vector.
\begin{equation*} e_i=\begin{pmatrix}0\\0\\\vdots\\0\\1\\0\\\vdots\\0\end{pmatrix}\in M_{n\times 1}(\K), \end{equation*}
where \(1\) occurs in the \(i\)-th row and all other entries are zero. We call \(e_i\) the \(i\)-th standard basis vector of \(M_{n\times 1}(\K)\).
We call the set of column vectors \(\{e_1,e_2,\ldots,e_n\}\) the standard basis.

Observation 1.7.3.

Suppose that \(v=\begin{pmatrix}\alpha_1\\\alpha_2\\\vdots\\\alpha_r\end{pmatrix}\in M_{r\times 1}(\K)\text{.}\) Then we have the following.
\begin{equation*} v=\alpha_1\,e_1+\alpha_2\, e_2+\cdots+\alpha_r\,e_r \end{equation*}

Observation 1.7.4.

Using Observation 1.7.3, if \(T\colon M_{n\times 1}(\K)\to M_{m\times 1}(\K)\) is a linear map then, for any \(v=\begin{pmatrix}\alpha_1\\\alpha_2\\\vdots\\\alpha_n\end{pmatrix}\) we get the following.
\begin{align*} T(v)\amp =T\begin{pmatrix}\alpha_1\\\alpha_2\\\vdots\\\alpha_n\end{pmatrix}\\ \amp =\alpha_1\,T(e_1)+\alpha_2\,T(e_2)+\cdots+\alpha_n\,T(e_n) \end{align*}
Thus, a linear map \(T\) is completely determined by column vectors \(T(e_1),T(e_2),\ldots,T(e_n)\).

Definition 1.7.5.

We keep notations of Observation 1.7.4. For a linear map \(T\) we associate a matrix, denoted by \([T]\text{,}\) whose \(i\)-th column vector is \(T(e_i)\text{,}\) for \(i\in\{1,2,\ldots,n\}\text{.}\) Thus, if
\begin{equation*} T(e_i)=\begin{pmatrix}a_{1i}\\a_{2i}\\\vdots\\a_{mi}\end{pmatrix}\in M_{m\times 1}(\K) \end{equation*}
then, associated matrix will be
\begin{equation*} [T]=\begin{pmatrix}a_{11}\amp a_{12}\amp\cdots\amp a_{1n}\\a_{21}\amp a_{22}\amp\cdots\amp a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{m1}\amp a_{m2}\amp\cdots\amp a_{mn}\\\end{pmatrix}\in M_{m\times n}(\K). \end{equation*}

Remark 1.7.6.

We remark that in Definition 1.7.5, the matrix associated with a linear map is always taken to be with respect to the standard basis.

Remark 1.7.7.

Let \(B\in M_{m\times n}(\K)\text{.}\) Consider the following map.
\begin{equation*} \ell_B\colon M_{n\times 1}(\K)\to M_{m\times 1}(\K)\quad\text{given by}\quad v\mapsto Bv \end{equation*}
This map is a linear map. Note that if
\begin{equation*} \begin{pmatrix}b_{11}\amp b_{12}\amp\cdots\amp b_{1n}\\b_{21}\amp b_{22}\amp\cdots\amp b_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\b_{m1}\amp b_{m2}\amp\cdots\amp b_{mn}\\\end{pmatrix} \end{equation*}
then, for any \(1\leq i\leq n\text{,}\) we get the following.
\begin{equation} \ell_B(e_i)=Be_i=\begin{pmatrix}b_{1i}\\b_{2i}\\\vdots\\b_{mi}\end{pmatrix}\tag{1.7.1} \end{equation}
Furthermore, the matrix of the linear map, \([\ell_B]\) is the following.
\begin{equation*} \left[\ell_B\right]=B \end{equation*}
Using Definition 1.7.5 and Remark 1.7.7 we get a one-one correspondence between the set of all \(m\times n\) matrices over \(\K\text{,}\) and the set of all linear maps from \(M_{n\times 1}(\K)\) to \(M_{m\times 1}(\K)\text{.}\) We note that in Definition 1.7.5, the matrix associated with a linear map is always taken to be with respect to the standard basis (Definition 1.7.2).
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