![[2005TelemanRepresentationTheory#^linear]] A **linear map** is a [[homomorphism]] between [[finite-dimensional|finite-dimensional vector spaces]], ie a [[function]] $T: V \twoheadrightarrow W$ that is [[additive]] and [[homogeneous]]: ![[additive]] ![[homogeneous]] See [[continuous linear map]] for the infinite-dimensional setting (where topological considerations emerge). Called **conjugate linear** if it instead has **conjugate** homogeneity. These coincide over real vector spaces Let $\mathrm{Lin}(V, W)$ be the set of all linear maps from $V$ to $W$. This is itself a vector space (ie another object of the category). Composition $\circ : \mathrm{Lin}(V, W) \times \mathrm{Lin}(U, V) \twoheadrightarrow \mathrm{Lin}(U, W)$ is [[bilinear form]]. ie $\begin{align*} (S + \lambda S') T &= ST + \lambda S' T \\ S(T + \lambda T') &= ST + \lambda ST' \end{align*}$ Linear maps are characterized by their effect on a [[linear basis]]. > [!NOTE]- proof > > ie let $v_{1:N}$ be a basis of $V$ and $w_{1:N}$ be some list in $W$. Then there exists a unique linear map $T : V \twoheadrightarrow W$ st $Tv_{n} = w_{n}$. > > Existence is trivially defined: $T(c^{n}v_{n}) := c^{n} w_{n}$. (This definition is fine since $v_{1:N}$ is a basis.) This is clearly linear. > > Uniqueness: Suppose $T \in \mathrm{Lin}(V, W)$ st $Tv_{n} = w_{n}$. Then $T(c^{n} v_{n}) = c^{n} w_{n}$ by linearity. $\dim \mathrm{Lin}(V,W) = (\dim V)(\dim W)$ > [!example] examples > > $ > \begin{array}{r|l} > \text{zero function} & 0 \in \mathcal L(V, W) \\ > \text{identity} & I \in \mathcal L(V, V) \\ > \text{differentiation} & D \in \mathcal L(\mathcal P(\mathbb R), \mathcal P(\mathbb R)) \\ > \text{integration} & T \in \mathcal L(\mathcal P(\mathbb R), \mathbb R) \\ > \text{multiplication by } x^2 & T \in \mathcal L(\mathcal P(\mathbb R), \mathcal P(\mathbb R)) \\ > \text{backward shift} & T(x_1, x_2, \cdots) = (x_2, x_3, \cdots) > \end{array} > $ ![[system of linear equations#^table]] # sources [[2015AxlerLinearAlgebraDone|Axler]] ch 3 3.3, 3.6, 3.7, 3.8, 3.9, 3.11, 3.60, 3.59, 3.40, 3.61