Affine Subspace¶

Definition¶

Let $V$ be a real vector space, $W\subseteq V$ a subspace and $\boldsymbol{v}\in V$ $V$ . The set

\{\boldsymbol{v}\}+W=\{\boldsymbol{v}+\boldsymbol{w}\colon\boldsymbol{w}\in W\}

is an affine subspace of $V$ . The subspace $W$ is its direction space, and the dimension of the affine subspace is defined as $\dim W$ . Some texts also count the empty set as affine; here an affine subspace is always nonempty.

Equivalently, a nonempty set $S\subseteq V$ is affine if and only if it contains every affine combination of its points: $\lambda\boldsymbol{x}+(1-\lambda)\boldsymbol{y}\in S$ for all $\boldsymbol{x},\boldsymbol{y}\in S$ and all $\lambda\in\mathbb{R}$ $R$ .

Interpretation¶

An affine subspace is a subspace shifted away from the origin: a point, a line, a plane or a higher-dimensional flat that need not pass through $\boldsymbol{0}$ . It is "flat" because the whole line through any two of its points stays inside it.

The line through (1,2)^T and (3,3)^T drawn as a copy of the dashed direction line W through the origin, shifted by v, with the point (-1,1)^T for lambda = 2 also on it.

The figure draws the line from the second example. The dashed red line is the direction space $W=\operatorname{span}\{(2,1)^{\top}\}$ through $\boldsymbol{0}$ ; the blue line is $\{\boldsymbol{v}\}+W$ with $\boldsymbol{v}=(1,2)^{\top}$ $(1, 2)^{⊤}$ , the same line shifted off the origin. It contains $(1,2)^{\top}$ and $(3,3)^{\top}$ , and also the affine combination $2(1,2)^{\top}-(3,3)^{\top}=(-1,1)^{\top}$ with $\lambda=2$ $2$ , which lies outside the segment between them. Since $\boldsymbol{0}$ is not on the blue line, $\{\boldsymbol{v}\}+W$ is not a subspace.

The condition $\lambda\in\mathbb{R}$ matters. Requiring only $\lambda\in[0,1]$ gives a convex set, so a line segment is convex but not affine.

The most common source of affine subspaces is linear systems. A consistent system of linear equations $A\boldsymbol{x}=\boldsymbol{b}$ with $\boldsymbol{b}\neq\boldsymbol{0}$ has a solution set that is not a subspace (it misses $\boldsymbol{0}$ ), yet it is one particular solution plus the kernel $\operatorname{Null}(A)$ . Affine subspaces are exactly the nonempty level sets of affine maps.

Key Results¶

The direction space is unique: $W=\{\boldsymbol{x}-\boldsymbol{y}\colon\boldsymbol{x},\boldsymbol{y}\in S\}$ $S}$ . The base point is not unique: $\{\boldsymbol{v}\}+W=\{\boldsymbol{v}'\}+W$ if and only if $\boldsymbol{v}-\boldsymbol{v}'\in W$ $W$ .
An affine subspace $\{\boldsymbol{v}\}+W$ is a linear subspace if and only if $\boldsymbol{0}\in\{\boldsymbol{v}\}+W$ $W$ , that is, if and only if $\boldsymbol{v}\in W$ $W$ .
If $A\boldsymbol{x}=\boldsymbol{b}$ is consistent and $\boldsymbol{x}_p$ is any solution, its solution set is $\{\boldsymbol{x}_p\}+\operatorname{Null}(A)$ $Null (A)$ , of dimension $n-\operatorname{rank}A$ for an $m\times n$ matrix $A$ .
A nonempty intersection of affine subspaces is affine; the image of an affine subspace under a linear map is affine.
In $\mathbb{R}^n$ the affine subspaces of dimension $n-1$ are the hyperplanes $\{\boldsymbol{x}\colon\boldsymbol{a}^{\top}\boldsymbol{x}=c\}$ with $\boldsymbol{a}\neq\boldsymbol{0}$ $0$ .

Examples¶

A plane in $\mathbb{R}^3$ . The solution set of $x_1+x_2+x_3=1$ contains $\boldsymbol{x}_p=(1,0,0)^{\top}$ $(1, 0, 0)^{⊤}$ . The direction space is the null space of $\begin{pmatrix}1&1&1\end{pmatrix}$ , spanned by $(-1,1,0)^{\top}$ and $(-1,0,1)^{\top}$ . Hence

S=\{(1,0,0)^{\top}\} +\operatorname{span}\{(-1,1,0)^{\top},(-1,0,1)^{\top}\},

an affine subspace of dimension $2$ . It is not a subspace, since $0+0+0\neq1$ $1$ .

A line in $\mathbb{R}^2$ . The line through $(1,2)^{\top}$ and $(3,3)^{\top}$ is $\{(1,2)^{\top}\}+\operatorname{span}\{(2,1)^{\top}\}$ $span {(2, 1)^{⊤}}$ . The affine combination with $\lambda=2$ $2$ , $2(1,2)^{\top}-(3,3)^{\top}=(-1,1)^{\top}$ $(- 1, 1)^{⊤}$ , is the point with parameter $t=-1$ $- 1$ , so it lies on the line, as it must.

Non-examples. The union of two distinct lines through $\boldsymbol{0}$ and the unit circle are not affine: in each case the line through two suitable points leaves the set.

Prerequisites¶

Sources¶

Math V: Advanced Linear Algebra · Week 5

Aliases: affine set, flat, linear manifold, translated subspace, linear variety, affine linear subspace, translate of a subspace Keywords: {v} + W, v + W, direction space, affine combination, solution set of Ax = b, particular solution plus null space Symbols: element of, lambda, real numbers, subset

Last updated: 22 October 2026 19:04