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 $\lambda \in F$

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 $D(f)=f'$ , 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, $\Hom_F(V,W)$ 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 $\Ker(T)=\{v\in V \mid T(v) = 0\}$ is a subspace of $V$ .
• The image $\Im(T) = \{T(v) \mid v\in V\}$ 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 $\Ker(T)=\{0\}$ .
• If $v \in V$ then $T^{-1}(T(v)) = v + \Ker(T)$ .
• If $w\in \Im(T)$ 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, linear transformation