> [!NOTE] Definiton > Let $A$ be a [[Real Matrices|real matrix]] of order $m\times n.$ The left multiplication linear maps of $A$ is given by the [[Function|map]] $\begin{align}L_{A}:\; &\mathbb{R}^{n} \to \mathbb{R}^{m} \\&\underline{v} \mapsto A \underline{v}\end{align}$where $A\underline{v}$ denotes a [[Real Matrix Product|matrix product]]. # Properties By [[Left Multiplication Linear Map of Real Matrix is Linear Map]], $L_{A}$ is indeed linear: that is, for all $v,w\in \mathbb{R}^{n}$ and $\lambda,\mu\in \mathbb{R},$ $L_{A}(\lambda v+\mu w)=\lambda L_{A}(v)+\mu L_{A}(w).$ > [!Note] Theorem (Smith normal gives image & kernel dimensions) > Let $A\in \mathbb{F}^{m\times n}$. Let $EA$ and $AF$ denoted the [[Reduced Row Echelon Form for Real Matrix|RREF]] and *RCEF* of $A$ such that $E=E_{k}\dots E_{1}$ is a product of [[Elementary Row Operation is Equivalent to Pre-Multiplying by Elementary Matrix|elementary matrices]] $E_{i}\in \mathbb{F}^{m\times m}$ and $F=F_{1}\dots F_{l}$ is a product of elementary matrices $F_{j}\in \mathbb{F}^{nn}$, then > 1. $\ker L_{EA}=\ker L_{A}$ and $\dim \text{Im}\, L_{EA} = \dim \text{Im} \, L_{A}$. > 2. $\text{Im} \, L_{AF} = \text{Im} \,L_{A}$ and $\dim \ker L_{AF} = \dim \ker L_{A}$. > 3. $\dim \ker L_{EAF} = \dim \ker L_{A}$ and $\dim \text{Im} \, L_{EAF} = \dim \text{Im} \, L_{A}$. ^453fd6 >See [[Reduced Row Echelon Form for Real Matrix|proof]]. > [!NOTE] Definition (Linear Map corresponding to $A$ wrt the given bases of $V$ and $W$) > Let $V$ be an [[Vector spaces|FDVS]] with basis $v_{1},\dots,v_{n}$ and $W$ an *FDVS* with basis $w_{1},\dots,w_{m}$. If $A\in \mathbb{R}^{m\times n}$ is a matrix, there is a unique [[Linear maps#^467233|linear map]] $\varphi_{A}$ defined on the basis of $V$ by $\begin{array}{rcl}\varphi_A\colon V&\longrightarrow&W\\v_i&\mapsto&a_{1i}w_1+\ldots+a_{mi}w_m\end{array}$where the coefficients of $\varphi_{A}(v_{i})$ with respect to the basis of $W$ are the $i$th column of $A$. > >Equivalently, $\varphi_{A}(\lambda_{1} v_{1}+\dots +\lambda_{n}v_{n}) = \mu_{1} w_{1}+\dots+\mu_{m} w_{m}$ where $\begin{pmatrix}\mu_{1} \\ \vdots \\ \mu_{m} \end{pmatrix} =A \begin{pmatrix}\lambda_{1} \\ \vdots \\ \lambda_{n} \end{pmatrix} $ ^2b50ef >Proof. Follows from the fact that [[Linear maps#^a8ff22|linear maps are equal if they agree on basis(2)]]. We have stated where we want the basis elements to map to, and therefore the unique linear map exists. > [!Example] > If $V=\mathbb{R}^{n}$ and $W=\mathbb{R}^{m}$ and if we choose the standard basis for $V$ and the standard basis for $W$, then $\varphi_{A}=L_{A}$, left multiplication linear map of $A$. > >It is enough to check that they agree on a basis: $\varphi_A(\underline{e}_i)=a_{1i}\underline{e}_1^{\prime}+\ldots+a_{mi}\underline{e}_m^{\prime}=\begin{pmatrix}a_{1i}\\\vdots\\a_{mi}\end{pmatrix}=A\underline{e}_i=L_A(\underline{e}_i)$are both simply the $i$th column of $A$. # Examples - [[Plane Rotation Matrix]]. - [[Plane Reflection Matrix]]. - [[Orthogonal projection of vector to line in a plane]]. # Applications - [[Matrix representations of linear map]]