Linear Operator

linear-algebra

Definition

Linear Operator

A linear operator is a linear map $T:V\to W$ between vector spaces. This means that for all vectors $u,v\in V$ and all scalars $\lambda$, one has

$$T(u+v)=T(u)+T(v)$$

and

$$T(\lambda u)=\lambda T(u).$$

If $V=W$, then $T$ is called a linear operator on $V$. In finite-dimensional spaces, every linear operator can be represented by a matrix.

Example: The map $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ given by

$$T\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}2x+y\\ y\end{array}\right)$$

is linear, since it preserves vector addition and scalar multiplication.