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Linear Algebra: Companion notes

Abhay Soman

Section 4.2 Examples of linear transformations

We give examples of linear transformations. Most of the details are left to the reader.
Let \(V\) be a vector space over a field \(F\text{.}\) For \(\alpha\in F\) consider the map left multiplication by \(\alpha\text{,}\)
\begin{equation*} \ell_\alpha\colon V\to V\quad\text{given by}\quad v\mapsto\alpha v. \end{equation*}
This is an \(F\)-linear transformation.
A group homomorphism \(T\colon V\to V\) is a linear transformation if and only if for every \(\alpha\in F\text{,}\) maps \(T\) and \(\ell_\alpha\) commute with each other, i.e., \(T\circ\ell_\alpha=\ell_\alpha\circ T\text{.}\)
We may consider a map \(\ell\colon F\to\End_F(V)\) defined by \(\alpha\mapsto\ell_\alpha\text{.}\) This map is an \(F\)-linear transformation as well.
Let \(F\) be a field and let \(A\in M_{m\times n}(F)\text{.}\) The map
\begin{equation*} L_A\colon M_{n\times 1}(F)\to M_{m\times 1}(F)\quad\text{given by}\quad x\mapsto Ax \end{equation*}
is an \(F\)-linear transformation. If \(e_i=(\delta_{i1},\delta_{i2},\ldots,\delta_{in})\) then, \(Ae_i^t\) is the \(i\)-th column of \(A\text{.}\)
For a natural number \(m\) consider a vector space \(\mathcal{P}_m(F)\) of all polynomials in one variable of degree at most \(m\) over a field \(F\text{.}\) The formal derivative mapping
\begin{equation*} D_n\colon\mathcal{P}_n(F)\to\mathcal{P}_{n-1}(F)\quad\text{defined by}\quad\sum_{i=0}^{n}a_iX^i\mapsto\sum_{i=1}^{n}ia_iX^{i-1} \end{equation*}
is an \(F\)-linear transformation.
Consider the vector space \(\R^2\) over \(\R\) and let \(e_1=(1,0)\text{.}\) We define a hyperplane reflection \(\tau_{e_1}\colon\R^2\to\R^2\) by
\begin{equation*} (x,y)\mapsto(-x,y). \end{equation*}
Note that any vector on the \(Y\)-axis, say \((0,y)\) maps to itself under this map. On the other hand, any vector on the \(X\)-axis, say \((x,0)\) is mapped to \((-x,0)\text{.}\) So, the the map \(\tau_{e_1}\) is a reflection along \(Y\)-axis. Check that \(\tau_{e_1}\) is an \(F\)-linear transformation.
Let \(F\) be a field and let \(M_n(F)\) be the vector space of \(n\times n\) matrices over \(F\text{.}\) Consider \(F\) as a vector space over itself. The trace map
\begin{equation*} \tr\colon M_n(F)\to F \end{equation*}
is given by
\begin{equation*} \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_{n1}\amp a_{n2}\amp\cdots\amp a_{nn}\end{pmatrix}\mapsto a_{11}+a_{22}+\cdots+a_{nn}. \end{equation*}
Check that this is an \(F\)-linear transformation.
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