Linear Operator — Definition, Formula & Examples

A linear operator is a function that maps vectors to vectors while preserving vector addition and scalar multiplication. In finite dimensions, every linear operator can be represented by a matrix.

A mapping $T: V \to W$ between vector spaces $V$ and $W$ over a field $F$ is a linear operator if for all vectors $\mathbf{u}, \mathbf{v} \in V$ and all scalars $c \in F$ : (1) $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ (additivity) and (2) $T(c\mathbf{u}) = cT(\mathbf{u})$ (homogeneity). When $V = W$ , the term "linear operator" is sometimes reserved specifically for this case, while the general case is called a "linear transformation."

Key Formula

T(c_1\mathbf{u} + c_2\mathbf{v}) = c_1\,T(\mathbf{u}) + c_2\,T(\mathbf{v})

Where:

$T$ = The linear operator (function between vector spaces)
$\mathbf{u}, \mathbf{v}$ = Vectors in the domain
$c_1, c_2$ = Scalars from the underlying field

How It Works

To check whether a function is a linear operator, you verify the two defining properties: additivity and homogeneity. These can be combined into a single condition:

T(c_1\mathbf{u} + c_2\mathbf{v}) = c_1 T(\mathbf{u}) + c_2 T(\mathbf{v})

. In practice, once you choose bases for the domain and codomain, a linear operator corresponds to multiplication by a matrix

A

, so that

T(\mathbf{x}) = A\mathbf{x}

. This connection means that studying matrices and studying linear operators are essentially the same thing in finite dimensions.

Worked Example

Problem: Determine whether the function

T(\mathbf{x}) = A\mathbf{x}

, where

A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}

, is a linear operator. Then compute

T\begin{pmatrix} 4 \\ -1 \end{pmatrix}

Step 1: Matrix multiplication always satisfies additivity and homogeneity:

A(c_1\mathbf{u} + c_2\mathbf{v}) = c_1 A\mathbf{u} + c_2 A\mathbf{v}

. So

T

is a linear operator.

Step 2: Compute the output by multiplying the matrix by the vector.

T\begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} 2(4)+1(-1) \\ 0(4)+3(-1) \end{pmatrix} = \begin{pmatrix} 7 \\ -3 \end{pmatrix}

Answer:

T

is a linear operator, and

T\begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} 7 \\ -3 \end{pmatrix}

Why It Matters

Linear operators are the central objects of study in linear algebra and appear throughout differential equations, quantum mechanics, and computer graphics. Recognizing that a process is linear lets you represent it as a matrix, unlocking powerful computational tools like eigenvalue decomposition and Cramer's Rule for solving systems.

Common Mistakes

Mistake: Assuming any function of vectors is linear. For example,

T(\mathbf{x}) = \mathbf{x} + \begin{pmatrix} 1 \\ 0 \end{pmatrix}

looks simple but is not linear because

T(\mathbf{0}) \neq \mathbf{0}

Correction: A linear operator must map the zero vector to the zero vector. If

T(\mathbf{0}) \neq \mathbf{0}

, the function fails linearity immediately — this is a quick first check.