Range of a Linear Map¶

Definition¶

Let $\boldsymbol{A}\in\mathcal{L}(V,W)$ be a linear map. The range or image of $\boldsymbol{A}$ is $\operatorname{range}\boldsymbol{A}=\{\boldsymbol{w}\in W\colon\exists \boldsymbol{v}\in V, \boldsymbol{A}\boldsymbol{v}=\boldsymbol{w}\}=\{\boldsymbol{A}\boldsymbol{v}\colon\boldsymbol{v}\in V\}$ $V}$ .

Interpretation¶

The range is the part of the codomain that is actually reached by the map. For a matrix it answers the question for which right-hand sides $\boldsymbol{b}$ the equation $\boldsymbol{A}\boldsymbol{v}=\boldsymbol{b}$ has a solution.

The line through the origin spanned by (1, 2), the range of a matrix with dependent columns, with two vectors v and u mapped onto points of this line.

The matrix $\boldsymbol{A}=\begin{pmatrix}1&2\\2&4\end{pmatrix}$ sends $\boldsymbol{v}$ to $\boldsymbol{A}\boldsymbol{v}=v_1\begin{pmatrix}1\\2\end{pmatrix}+v_2\begin{pmatrix}2\\4\end{pmatrix}=(v_1+2v_2)\begin{pmatrix}1\\2\end{pmatrix}$ $2 v_{2}) (12)$ . Both drawn vectors $\boldsymbol{v}$ and $\boldsymbol{u}$ land on the line through $(1,2)$ , and so does every other output: the range is the line spanned by the columns, not the whole plane.

Every current output $\boldsymbol{M}_t\boldsymbol{x}$ lies in the shaded range of $\boldsymbol{M}_t=(1-t)\boldsymbol{I}+t\boldsymbol{A}$ $t A$ . Its range is the plane, a line or the origin according to its rank. Move $t$ from the identity at $0$ to the target $\boldsymbol{A}$ at $1$ .

The same computation works for any matrix: $\boldsymbol{A}\boldsymbol{v}$ is the linear combination of the columns of $\boldsymbol{A}$ with weights $v_1,\ldots,v_n$ . Hence the range of $\boldsymbol{v}\mapsto\boldsymbol{A}\boldsymbol{v}$ is the span of the columns, also called the column space; $\boldsymbol{A}\boldsymbol{v}=\boldsymbol{b}$ is solvable exactly when $\boldsymbol{b}$ lies in it, and its dimension is the rank of $\boldsymbol{A}$ .

The word is used as for any function in Math I: the set of values actually taken, as opposed to the codomain, which only says where values may lie. A linear map is surjective if and only if its range is all of $W$ , and by the fundamental theorem of linear maps a kernel of larger dimension leaves a range of smaller dimension.

A common slip is to take the span of the rows instead of the columns. For $\begin{pmatrix}1&1\\0&0\end{pmatrix}$ the rows span the line through $(1,1)$ , but every output $(v_1+v_2,0)$ lies on the $x_1$ -axis.

Key Results¶

The range of a linear map is a subspace of the codomain.
A linear map is surjective exactly when its range equals the codomain.

Examples¶

For $\boldsymbol{A}(x,y)=(x,0)$ on $\mathbb{R}^2$ , the range is the $x$ -axis.
The range of the zero map is $\{\boldsymbol{0}\}$ .

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 4
Math V: Advanced Linear Algebra · Week 5 (revisited)

Aliases: image of a linear map Keywords: range, image, column space Symbols: element of, real numbers, there exists

Last updated: 27 October 2026 11:52