Subspace¶

Definition¶

Let $V$ be a vector space over a field $\mathbb{F}$ . A subspace or linear subspace of $V$ is a nonempty subset $U\subseteq V$ that is itself a vector space using the same vector addition, scalar multiplication, and zero vector as $V$ .

Equivalently, $U$ is a subspace if $U$ is nonempty and for all $\boldsymbol{u}_1,\boldsymbol{u}_2\in U$ and all $\alpha,\beta\in\mathbb{F}$ $F$ ,

\alpha \boldsymbol{u}_1+\beta \boldsymbol{u}_2\in U.

Interpretation¶

A subspace is a smaller vector space sitting inside a larger one, such as a line or a plane through the origin inside $\mathbb{R}^3$ . The operations are not new: addition and scalar multiplication are inherited from the surrounding vector space, so the only question is whether they stay inside the subset.

Two panels: a plane through the origin, closed under adding vectors such as u1 and u2, which is a subspace, and a shifted plane that misses the origin and does not contain 2u, which is not.

The left panel shows a plane through the origin: adding $\boldsymbol{u}_1$ and $\boldsymbol{u}_2$ gives $\boldsymbol{u}_1+\boldsymbol{u}_2$ $u_{2}$ , which again lies in the plane, and so does every multiple of a vector in it. The right panel shows a shifted plane $U$ that misses $\boldsymbol{0}$ . It contains $\boldsymbol{u}$ , but not $2\boldsymbol{u}$ , so scaling leaves $U$ and $U$ is not a subspace.

This is why closure matters. The single condition $\alpha\boldsymbol{u}_1+\beta\boldsymbol{u}_2\in U$ covers both operations: $\alpha=\beta=1$ gives closure under addition, and $\beta=0$ gives closure under scalar multiplication. The other vector space rules, such as associativity, need no check, because they already hold in $V$ .

The zero vector is forced: since $U$ is nonempty, taking $\alpha=\beta=0$ gives $\boldsymbol{0}\in U$ $U$ . This quick test rules out many affine-looking sets, such as shifted lines or planes not passing through the origin, even though they look geometrically similar. Containing $\boldsymbol{0}$ is necessary but not sufficient: the two coordinate axes of $\mathbb{R}^2$ together contain $\boldsymbol{0}$ , but $(1,0)+(0,1)=(1,1)$ lies on neither axis. Typical subspaces are the span of some vectors and the solutions of an equation such as $x+y=0$ $0$ .

Key Results¶

Subspace test: a nonempty subset $U\subseteq V$ is a subspace if and only if it is closed under all linear combinations of two of its vectors.
Equivalently, $U$ is a subspace if it contains the zero vector and is closed under vector addition and scalar multiplication.
The two trivial subspaces of $V$ are $\{\boldsymbol{0}\}$ and $V$ itself.
The kernel of a linear map is always a subspace of its domain, and the range is always a subspace of its codomain.
Every subspace of a finite-dimensional vector space is finite-dimensional and has dimension at most that of the original space.

Examples¶

The line $\{(x,0)\colon x\in\mathbb{R}\}$ is a subspace of $\mathbb{R}^2$ because it contains $(0,0)$ and is closed under linear combinations.
The set $\{(x,y)\colon x+y=1\}$ is not a subspace of $\mathbb{R}^2$ because it does not contain the zero vector.
The singleton set $\{\boldsymbol{0}\}$ and the full space $V$ are the trivial subspaces of $V$ .
A hyperplane through the origin is a subspace; a parallel shifted hyperplane is generally not.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 2
Math V: Advanced Linear Algebra · Week 5 (revisited)

Aliases: linear subspace, vector subspace Keywords: trivial subspace Symbols: alpha, beta, element of, real numbers, subset

Last updated: 20 October 2026 03:02