Linear Operator¶

Definition¶

A linear operator is a linear map from a vector space to itself, so it has the form $\boldsymbol{A}\colon V\to V$$V$.

Interpretation¶

A linear operator is a linear map that stays inside one space: input and output are vectors of the same vector space $V$. This makes repeated application meaningful, because the operator can be applied again to its own output.

The illustrated operator rotates every vector of the plane by $30^\circ$ about the origin and is represented by $\boldsymbol{A}=\begin{pmatrix}\cos30^\circ&-\sin30^\circ\\\sin30^\circ&\cos30^\circ\end{pmatrix}$$\left(\begin{array}{cc}\cos 30^{\circ }& -\sin 30^{\circ }\\ \sin 30^{\circ }& \cos 30^{\circ }\end{array}\right)$. The triangle $S$ and the vector $\boldsymbol{v}$ are moved to $\boldsymbol{A}(S)$ and $\boldsymbol{A}\boldsymbol{v}$ in the same plane. Its domain and codomain are the same space, so powers represent repeated application: $\boldsymbol{A}^2$ rotates by $60^\circ$, and $\boldsymbol{A}^{12}$ rotates by $360^\circ$, which is the identity.

The difference with a general linear map lies in the codomain. For a general $\boldsymbol{A}\colon V\to W$$W$, the composition $\boldsymbol{A}\boldsymbol{A}$ need not be defined: the output must lie in the domain $V$ to be used again. An operator guarantees this compatibility. A map with $W\subseteq V$ can also be applied repeatedly, even when $W\neq V$$V$. In matrix terms, an operator on $\mathbb{R}^n$ is given by a square $n\times n$ matrix; ordinary matrix powers require square matrices.

The same formula can give an operator or not, depending on the codomain: $(x_1,x_2)\mapsto(x_1,0)$ is an operator on $\mathbb{R}^2$, while $(x_1,x_2)\mapsto x_1$$x_1$, from $\mathbb{R}^2$ to $\mathbb{R}$, is not. Questions such as which vectors an operator only stretches, or whether its effect can be undone, are asked for operators and return later as eigenvectors and inverses.

Key Results¶

• The set of linear operators on $V$ is denoted $\mathcal{L}(V)=\mathcal{L}(V,V)$$\mathcal{L}(V,V)$.
• Eigenvalues, determinants, inverses, and diagonalization are concepts for linear operators or their square matrix representations.

Examples¶

• The identity map $\boldsymbol{I}(\boldsymbol{v})=\boldsymbol{v}$ is a linear operator on $V$.
• The derivative $\boldsymbol{D}\colon p\mapsto p'$ is a linear operator on the vector space of polynomials.

Prerequisites¶

• Linear Map
• Vector Space

Related Concepts¶

• Matrix
• Product of Linear Maps
• Idempotent Matrix

Sources¶

• Math II: Linear Algebra · Week 3

Aliases: linear endomorphism Keywords: endomorphism Symbols: real numbers, subset

Last updated: 27 October 2026 11:52