[[Types]]:: *[[Types]]* [[Examples]]:: *[[Examples]]* [[Nonexamples]]:: *[[Nonexamples]]* [[Constructions]]:: *[[Constructions|Used in the construction of...]]* [[Generalizations]]:: *[[Generalizations]]* \ [[Properties]]:: *[[Properties]]* [[Sufficiencies]]:: *[[Sufficiencies]]* [[Equivalences]]:: *[[Equivalences]]* [[Justifications and Intuition]]:: *[[Justifications and Intuition]]* - Let $V,W$ be [[submodule generated by a subset|finitely generated]] $R$-[[module|modules]] for $R$ an [[integral domain]] (e.g., [[vector space#Finite-Dimensional Vector Space|finite-dimensional vector spaces]]); - Let $T:V \to W$ be a [[linear map]] (e.g., a [[linear map]]), $T \in \text{Hom}_{R\text{-}\mathsf{Mod}}(V,W)$; - Let $A=\lb v_i \rb_{i=1}^n \and B=\lb w_i \rb_{i=1}^m$ be [[basis|bases]] for $V$ and $W$ respectively. From [[module is free iff admits basis|free module iff has basis]] and/or [[the rank theorem for free modules]], we know that specifying $A$ and $B$ amounts to specifying identifications $V \cong R^{\oplus A}$ and $W \cong R^{\oplus B}$. Any matrix definition, strictly speaking, applies only to maps between such '[[direct sum of modules|direct sum]] [[free module|coordinate spaces]]')... the phrase 'with respect to the bases $A$ and $B below indicates that $T$ should be interpreted as a map from $R^{\oplus A} \cong V$ to $R^{\oplus B} \cong W$. > [!definition] Definition. ([[matrix]]) > The **matrix** $\MM(T)$ of $T$ *with respect to the [[basis|bases]] $A$ and $B$* is the $m$-by-$n$ array of elements for which the $k^{th}$ column is the scalars needed to write $Tv_k$ as a [[linear combination]] of $(w_1,\dots,w_m)$. That is, the $k^{th}$ column is the [[coordinate vector]] of $Tv_k$ with respect to the [[basis]] $w_1, \dots, w_m$. $\MM(T)=\begin{align} &\begin{matrix} & v_1 & \dots & v_k & \dots & v_n \end{matrix} \\ &\begin{matrix} w_1 \\ \vdots \\ w_m \end{matrix} \begin{pmatrix} & & A_{1,k} & \\ & & \vdots & \\ & &A_{m,k} & &\end{pmatrix} \\ =& \begin{pmatrix} \vert & \vert & \vert \\ [Tv_1]_{\{w_j\}_{1}^m} & \dots & [Tv_n]_{\{w_j\}_{1}^m} \\ \vert & \vert & \vert \end{pmatrix} \\ =& \begin{pmatrix} \vert & \vert & \vert \\ \MM(Tv_1) & \dots & \MM(Tv_n) \\ \vert & \vert & \vert \end{pmatrix}. \end{align} $\ If the [[basis|bases]] are not clear from context, then the notation $\mathcal{M}(T; A, B)=\MM\left(T; \lb v_i \rb_{i=1}^n, \lb w_i \rb_{i=1}^m \right)$ is used. Otherwise we just write $\mathcal{M}(T)$ (if we are dealing with [[free module|coordinate spaces]] the [[basis|bases]] always default to [[free module|standard basis]]). > > [!definition] The matrix ring. > Matrices over a [[ring]] $R$ themselves form a [[ring]] under addition and multiplication (an $R$-[[algebra]], actually, once we added in scalar multiplication). ^definition > [!NOTE] Remark. > This definition states that the $k^{th}$ column of $\MM(T)$ consists of the [[scalar]]s needed to write $Tv_k$ as a [[linear combination]] of $w_1, \dots, w_m$. Likewise, we can obtain $Tv_k$ by taking a [[linear combination]] of the $ws, weighted by the corresponding elements in the $k^{th}$ column. $Tv_k = \sum_{j=1}^m A_{j,k}w_j.$ > [!equivalence] > There is a second (equivalent) way to define the matrix of a [[linear map]]/[[linear map]] $T$, offered e.g. in Math 217 and in Aluffi. It is presented in two steps, which Axler's definition above packages into a single step. > > **Step 1.** The **matrix** $\mathcal{M}(T)$ of a [[linear map]]/[[linear map]] $T$ between ([[submodule generated by a subset|finitely generated]]) *[[free module|coordinate spaces]]* $R^{\oplus A}$ and $R^{\oplus B}$ has as columns the 'standard basis'/'indicator' images under $T$:$ \begin{align} > \mathcal{M}(T)_{:, \ell} := T\big( (0, \dots, 0,\overbrace{1}^{\ell^{th}\text{ slot}} ,0,\dots,0) \big). > \end{align}$ > > **Step 2.** Step 1 doesn't quite make sense for defining the matrix $\mathcal{M}(T)$ of a [[linear map]]/[[linear map]] between *arbitrary* ([[submodule generated by a subset|finitely generated]]) [[free module|free modules]]/[[vector space|vector spaces]] $T:F \to G$: a priori, we don't *have* a 'basis' to work with, much less a 'preferred' basis like the term 'standard basis' suggests we had in step 1. > > The remedy, of course, is to [[coordinate isomorphism|identify]] $F \cong R^{\oplus A}$ and $G \cong R^{\oplus B}$ for [[basis|bases]] $A=\{ v_{i} \}_{i=1}^{n},B=\{ w_{i} \}_{i=1}^{m}$ of $F,G$. There is no canonical way to do this, so it is crucial that, once we have made such an identification ('chosen a [[coordinate isomorphism]]'), we thereon speak of the 'matrix of $T$ *with respect to $A$ and $B$*'. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRACUA9YAHR4jTM4AAgCCAXxDjS6TLnyEUARnJVajFmwBiUmSAzY8BIgCZV1es1aIQAcV2zDCogGZz6q2y69+gpiIAhSXE1GCgAc3giUAAzACcIAFskMhAcCCQVD00bPno4tAALLBBqBjoAIxgGAAU5I0UQBhgYnAcQeKSU6nTM6ViE5MRU3sQsyxyQPmxkssrquqdjGziscMK2-o7BpDc0jMQzbOspnjRsUqb52vrnFbWN9s6hvdGj8qqbpcbm1suJk7TLDJLbPTI9A5HAFsAAqTx2wwhu2oVTAUF2qWhuTO2G4AFolOJhHwAMZYOIk4Qw4k8MkUmn5IolELiIA > \begin{tikzcd} > R^{\oplus A} \arrow[r, "\varphi"] \arrow[r] \arrow[r, "\sim"'] \arrow[rrr, "\psi^{-1} \circ T \circ \varphi", bend left] & F \arrow[r, "T"] & G & R^{\oplus B} \arrow[l, "\psi"'] \arrow[l, "\sim"] > \end{tikzcd} > \end{document} > ``` > > > With notation per the diagram, this matrix is defined as the matrix (ala step 1) of the map $\psi ^{-1} \circ T \circ \varphi$. > ^equivalence > > > > > > >> [!justification] > > >To see this definition agrees with the original, we have to recall the definitions of the [[coordinate isomorphism|coordinate isomorphisms]] $\varphi$ and $\psi$: $\varphi$ is determined by its identification of each 'standard basis vector' $(0,\dots,0,\overbrace{1}^{k^{th} \text{ slot}},0,\dots,0)\in R^{\oplus A}$ with $v_{k} \in F$. $\psi ^{-1}$ is determined by its identification of each $w_{i} \in G$ with $(0,\dots,0,\overbrace{1}^{i^{th} \text{ slot}},0,\dots,0) \in R^{\oplus B}$. Now, $\begin{align}\psi ^{-1}\big( T (\varphi(0, \dots, 0, \overbrace{1}^{k^{th} \text{ slot}}, 0, .., 0)) \big) = & \psi ^{-1}(Tv_{\ell}) \\ > >= & \psi ^{-1}\left( \sum_{i=1}^{m} a_{i} w_{i} \right) , \text{ for some } a_{i} \\ > >= & \sum_{i=1} a_{i} \psi ^{-1}(w_{i}) \\ > >= & \sum_{i=1} (0, \dots, 0, \overbrace{a_{i}}^{i^{th} \text{ slot}}, 0, \dots, 0) \\= & (a_{1}, \dots, a_{m}) \end{align}$ where the $a_{i}$ are precisely the scalars needed to write $Tv_{\ell}$ as a [[linear combination]] of the $w_{i}$. > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```