# linear transformation

Let $V$ and $W$ be vector spaces over the same field $F$. A linear transformation is a function $T\colon V\to W$ such that:

• $T(v+w)=T(v)+T(w)$ for all $v,w\in V$

• $T(\lambda v)=\lambda T(v)$ for all $v\in V$, and $\lambda\in F$

The set of all linear maps $V\to W$ is denoted by $\operatorname{Hom}_{F}(V,W)$ or $\mathscr{L}(V,W)$.

Examples:

• Let $V=\mathbb{R}^{n}$ and $W=\mathbb{R}^{m}$ and $A$ is any $m\times n$ matrix. Then the function $L_{A}:V\to W$ defined by $L_{A}(v)=Av$, the multiplication of matrix $A$ and the vector $v$ (considered as an $n\times 1$ matrix), is a linear transformation.

• Let $V$ be the space of all differentiable functions over $\mathbb{R}$ and $W$ the space of all continuous functions over $\mathbb{R}$. Then $D:V\to W$ defined by $D(f)=f^{\prime}$, the derivative of $f$, is a linear transformation.

Properties:

• $T(0)=0$.

• If $S$ and $T$ are linear transformations from $V$ to $W$, and $k\in F$, then so are $S+T$ and $kT$. As a result, $\operatorname{Hom}_{F}(V,W)$ is a vector space over F.

• If $G\colon W\to U$ is a linear transformations then $G\circ T\colon V\to U$ is also a linear transformation.

• The kernel (http://planetmath.org/KernelOfALinearTransformation) $\operatorname{Ker}(T)=\{v\in V\mid T(v)=0\}$ is a subspace of $V$.

• The image (http://planetmath.org/ImageOfALinearTransformation) $\operatorname{Im}(T)=\{T(v)\mid v\in V\}$ is a subspace of $W$.

• The inverse image $T^{-1}(w)$ is a subspace if and only if $w=0$.

• A linear transformation is injective if and only if $\operatorname{Ker}(T)=\{0\}$.

• If $v\in V$ then $T^{-1}(T(v))=v+\operatorname{Ker}(T)$.

• If $w\in\operatorname{Im}(T)$ then $T(T^{-1}(w))=\{w\}$.

Remark. A linear transformation $T:V\to W$ such that $W=V$ is called a linear operator, and a linear functional when $W=F$.