26 - Usage of Linear Maps

Left Multiplication Map

For $m,n\ge 1$ with $A\in {\mathcal{M}}_{m\times n}(\mathbb{R})$
Left multiplication map $L_A:{\mathbb{R}}_{\mathrm{col}}^n$ by

$$L_A(\mathbf{v})=A\mathbf{v}\text{ for }\mathbf{v}\in {\mathbb{R}}_{\mathrm{col}}^n$$

Right Multiplication Map

For $m,n\ge 1$ with $A\in {\mathcal{M}}_{m\times n}(\mathbb{R})$
Left multiplication map $L_A:{\mathbb{R}}_{\mathrm{col}}^n$ by

$$L_A(\mathbf{v})=A\mathbf{v}\text{ for }\mathbf{v}\in {\mathbb{R}}_{\mathrm{col}}^n$$

Multiplication Maps with Matrices

Take $m,n,p\ge 1$ with $A\in {\mathcal{M}}_{m\times n}(\mathbb{R})$
The left multiplication map ${\mathcal{M}}_{n\times p}(\mathbb{R})\to {\mathcal{M}}_{m\times p}(\mathbb{R})$ sends $X$ to $AX$ is also a linear map

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Useful example of a Linear Maps

Let ${\mathbb{R}}_n[x]$ be the vector space of polynomials of degree at most $n$
Define $D:{\mathbb{R}}_n[x]\to {\mathbb{R}}_n[x]$ by $p(x)\mapsto p^{\prime }(x)$
that is

$$D(a_n{x}^n+\cdots +a_{1}x+a_0)=n{a}_n{x}^{n-1}+\cdots +a_1$$

It is a linear map from ${\mathbb{R}}_n[x]$ to ${\mathbb{R}}_n[x]$
Could also be a linear map from ${\mathbb{R}}_n[x]$ to ${\mathbb{R}}_{n-1}[x]$

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Row Space and Column Space corollary

Given a matrix $A$
Image of $L_A$ is $\text{Col}(A)$, the column space of $A$
Image of $R_A$ is $\text{Row}(A)$, the row space of $A$

Proof

The Canonical Basis ${\mathbf{e}}_1,\cdots ,{\mathbf{e}}_m$ spans ${\mathbb{R}}^m$
Hence the rows of $A$, ${\mathbf{r}}_1={\mathbf{e}}_{1}A,\cdots ,{\mathbf{r}}_m={\mathbf{e}}_{m}A$ spans $\mathrm{Im}{R}_A$
Hence $\mathrm{Im}{R}_A\subseteq \mathrm{Row}(A)$

Conversely if $\mathbf{v}\in \mathrm{Row}(A)$ then

$$\mathbf{v}={\alpha }_1{\mathbf{r}}_1+\cdots +{\alpha }_m{\mathbf{r}}_m=({\alpha }_1{\mathbf{e}}_{1+\cdots +{\alpha }_m{\mathbf{e}}_m})\in \mathrm{Im}{R}_A$$

Similarly then $\mathrm{Im}{L}_A=\mathrm{Col}(A)$