---- > [!definition] Definition. ([[linear map]]) > Let $R$ be a [[ring]]. A **homomorphism of $R$-modules** $M$ and $N$ is a [[group homomorphism|homomorphism]] $\varphi:M \to N$ of [[abelian group|abelian groups]] that is compatible with the [[module]] structure: for all $m, m_{1},m_{2} \in M$, $r \in R$, $\begin{align} \varphi(m_{1} + m_{2}) = & \varphi(m_{1}) + \varphi(m_{2}); \\ \varphi(rm) = & r \varphi(m) . \end{align}$ These conditions are often called 'additivity' and 'homogeneity'. > >Note that $\varphi(M)$ is a [[submodule]] of $N$. ^definition > [!specialization] > For $R=k$ a [[field]] this is precisely the definition of [[linear map]]. ^specialization > [!basicexample] > - A $\mathbb{Z}$-module [[group homomorphism|homomorphism]] is just a [[group homomorphism|homomorphism]] of [[abelian group|abelian groups]]. Indeed, recalling [[every abelian group is a Z-module, in exactly one way]], the condition $\varphi(rm)=r\varphi(m)$ specializes in this case to $\varphi(\overbrace{m+\dots + m}^{r \text{ times}})=\overbrace{\varphi(m) + \dots + \varphi(m)}^{r \text{ times}}$ which is necessarily and sufficiently satisfied for $\varphi$ just a [[group homomorphism]]. Hence the [[category]] $\mathbb{Z}$-$\mathsf{Mod}$ is merely $\mathsf{Ab}$. > - [[linear map|Linear maps]], when $R$ is a [[field]]. ^basic-example ---- #### ---- #### 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 > ``` Examples:: [[linear map#^f85587|basic examples]], [[derivative|differentiation]], [[linearity of the integral|integration]], Nonexamples:: *[[Nonexamples]]* Constructions:: [[vector space of operators on a vector space]], [[vector space of linear maps between two vector spaces]] Generalizations:: [[multilinearity]], Justifications and Intuition:: *[[Justifications and Intuition]]* ---- Merge: [[linear map]] Let $V$ and $W$ be [[vector space]]s over a [[field|field]] $\ff$. > [!definition] Definition. ([[linear map]]) > A **linear map** from $V$ to $W$ is a function $T: V \to W$ with the following properties: \ **Additivity.** $T(u+v)=T(u)+ T(v)$ for all $u,v \in V$. \ **Homogeneity.** $T(\lambda v) = \lambda T(v)$ for all $\lambda \in \ff$. ---- > [!basicexample]- Basic Examples. > - $0:V \to W$: which takes each element in $V$ to the additive identity $0$ in $W$. > - The **Identity Map** $I$ :$V\to V$ that takes each $v \in V$ to itself. > - Define $D \in \hom(\PP(\rr), \PP(\rr))$ by $Dp=p'$. > - Define $T \in \hom(\PP(\rr), \rr)$ by $Tp = \int_{0}^1 p(x) \ \d x$. See [[linearity of the integral]]. > - Define $T \in \hom(\PP(\rr), \PP(\rr))$ by $(Tp)(x) = x^2 p(x).$ ---- #### 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 > ```