Linear Map¶

Definition¶

Let $\mathbb{F}$ be a field, and let $V$ and $W$ be vector spaces over $\mathbb{F}$. A linear map $\boldsymbol{A}\colon V\to W$ is a function satisfying additivity $\boldsymbol{A}(\boldsymbol{u}+\boldsymbol{v})=\boldsymbol{A}\boldsymbol{u}+\boldsymbol{A}\boldsymbol{v}$ and homogeneity $\boldsymbol{A}(\lambda \boldsymbol{u})=\lambda \boldsymbol{A}\boldsymbol{u}$ for all $\boldsymbol{u},\boldsymbol{v}\in V$ and $\lambda\in\mathbb{F}$$\mathbb{F}$. The set of all linear maps from $V$ to $W$ is denoted $\mathcal{L}(V,W)$.

Basic linear maps and operations¶

Two canonical examples are the zero map $\boldsymbol{0}\colon V\to W$$W$, defined by $\boldsymbol{0}(\boldsymbol{v})=\boldsymbol{0}_W$$0_W$, and the identity map $\boldsymbol{I}\colon V\to V$$V$, defined by $\boldsymbol{I}(\boldsymbol{v})=\boldsymbol{v}$$\mathbf{v}$. If $\boldsymbol{A},\boldsymbol{B}\in\mathcal{L}(V,W)$ and $\lambda\in\mathbb{F}$$\mathbb{F}$, then the sum and scalar multiple of linear maps are defined by

These operations make $\mathcal{L}(V,W)$ itself a vector space.

Interpretation¶

A linear map preserves the linear structure of a vector space: it respects vector addition and scalar multiplication. It does not matter whether vectors are first combined and then mapped, or first mapped and then combined: $\boldsymbol{A}(\lambda\boldsymbol{u}+\mu\boldsymbol{v})=\lambda\boldsymbol{A}\boldsymbol{u}+\mu\boldsymbol{A}\boldsymbol{v}$$\mu \mathbf{A}\mathbf{v}$.

The square grid of $V$, with $\boldsymbol{e}_1$, $\boldsymbol{e}_2$ and $\boldsymbol{v}=\boldsymbol{e}_1+\boldsymbol{e}_2$${\mathbf{e}}_2$, is mapped to a slanted grid. By additivity, $\boldsymbol{A}\boldsymbol{v}=\boldsymbol{A}\boldsymbol{e}_1+\boldsymbol{A}\boldsymbol{e}_2$$\mathbf{A}{\mathbf{e}}_2$: the diagonal of the unit square goes to the diagonal of the parallelogram formed by $\boldsymbol{A}\boldsymbol{e}_1$ and $\boldsymbol{A}\boldsymbol{e}_2$. The grid lines stay straight, parallel and evenly spaced, and $\boldsymbol{0}$ stays at $\boldsymbol{0}$. A linear map is determined by its images of a basis; here $\boldsymbol{A}\boldsymbol{e}_1$ and $\boldsymbol{A}\boldsymbol{e}_2$ are the matrix columns.

Every linear map sends $\boldsymbol{0}$ to $\boldsymbol{0}$, because $\boldsymbol{A}\boldsymbol{0}=\boldsymbol{A}(0\cdot\boldsymbol{0})=0\cdot\boldsymbol{A}\boldsymbol{0}=\boldsymbol{0}$$0$. This gives a quick test: $g(x,y,z)=x-y+2z+1$ from the Examples fails it. Fixing the origin is not enough, though: $x\mapsto x^2$ on $\mathbb{R}$ sends $0$ to $0$, but $(1+1)^2=4\neq1^2+1^2$$1^2$, so additivity fails.

The two conditions are differentiation rules already known from Math I: $(f+g)'=f'+g'$ and $(cf)'=cf'$$c{f}^{\prime }$, which is why differentiation is a linear map. A common slip comes from the everyday name "linear function" for $f(x)=ax+b$$b$: it is a linear map only when $b=0$$0$; for $b\neq0$ its graph is a line that misses the origin.

Key Results¶

• A linear map is completely determined by what it does to a basis of its domain.
• The set $\mathcal{L}(V,W)$ of all linear maps from $V$ to $W$ is itself a vector space.
• A function $\boldsymbol{f}\colon\mathbb{R}^n\to\mathbb{R}^k$ is linear if and only if $\boldsymbol{f}(\boldsymbol{x})=\boldsymbol{A}\boldsymbol{x}$ for some $k\times n$ matrix $\boldsymbol{A}$, and then $\boldsymbol{A}$ is unique: its $j$-th column is $\boldsymbol{f}(\boldsymbol{e}_j)$. In particular a linear map sends $\boldsymbol{0}$ to $\boldsymbol{0}$, so a function $\boldsymbol{x}\mapsto \boldsymbol{A}\boldsymbol{x}+\boldsymbol{b}$ with $\boldsymbol{b}\ne\boldsymbol{0}$ is not linear but affine.
• Linear map and linear transformation are synonyms. A linear map from a space to itself is also called a linear operator.

Examples¶

• The zero map $\boldsymbol{0}(\boldsymbol{v})=\boldsymbol{0}$ is linear.
• The derivative map $\boldsymbol{D}(p)=p'$ on polynomials is linear.
• $f(x,y)=x+y$ is a linear map $\mathbb{R}^2\to\mathbb{R}$$\mathbb{R}$; $g(x,y,z)=x-y+2z+1$ is not, since $g(0,0,0)=1\ne0$$0$.

Prerequisites¶

• Vector Space
• Field
• Function

Related Concepts¶

• Matrix

Sources¶

• Math II: Linear Algebra · Week 3
• Math III: Analysis · Lecture 7

Aliases: linear transformation, linear mapping, vector space homomorphism Keywords: zero map, identity map, sum of linear maps, scalar multiple of a linear map, standard matrix, matrix of a linear map Symbols: element of, lambda, mu, real numbers

Last updated: 20 October 2026 03:02