linear transformation

Let $V$ and $W$ be vector spaces over the same field $F$. A linear transformation is a function $T:V\to W$ 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 ${\mathrm{Hom}}_F(V,W)$ or $ℒ(V,W)$.

Examples:

• Let $V={ℝ}^n$ and $W={ℝ}^m$ 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 $ℝ$ and $W$ the space of all continuous functions over $ℝ$. 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, ${\mathrm{Hom}}_F(V,W)$ is a vector space over F.

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

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

• The image (http://planetmath.org/ImageOfALinearTransformation) $\mathrm{Im}(T)=\{T(v)\mid v\in V\}$ 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 $\mathrm{Ker}(T)=\{0\}$.

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

• If $w\in \mathrm{Im}(T)$ 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 $W=F$.

See also:

• 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