# 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 $W=F$.

• Wikipedia, http://www.wikipedia.org/wiki/Linear_transformationlinear transformation

 Title linear transformation Canonical name LinearTransformation Date of creation 2013-03-22 11:56:41 Last modified on 2013-03-22 11:56:41 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 24 Author CWoo (3771) Entry type Definition Classification msc 15A04 Synonym linear map Synonym vector space homomorphism Synonym linear mapping Related topic Matrix Related topic InvariantSubspace Related topic DualHomomorphism Related topic KernelOfALinearTransformation Related topic EigenvalueOfALinearOperator Related topic NilpotentTransformation Related topic AffineTransformation Related topic SubLinear Related topic MatrixRepresentationOfALinearTransformation Defines linear operator