Linear Subspace

Not every subset of a vector space is itself a vector space. A random subset can fail to contain the zero vector, or it can “escape” the subset when you add two of its elements. Linear subspaces are the subsets that preserve the vector space structure — and they appear naturally everywhere in linear algebra: as kernels, as images, as solution sets of homogeneous systems.

Definition

Let $V$ be a vector space over a field $F$. A non-empty subset $W \subseteq V$ is a linear subspace (also simply called a subspace) of $V$ if it satisfies all three of the following conditions:

1. $\mathbf{0} \in W$  (the zero vector belongs to $W$)
2. $u, v \in W \implies u + v \in W$  (closed under addition)
3. $u \in W,\ c \in F \implies cu \in W$  (closed under scalar multiplication)

When these hold, $W$ is a vector space in its own right, using the same operations as $V$. All the vector space axioms — associativity, distributivity, etc. — are inherited automatically from $V$, because $W$‘s elements and operations are simply those of $V$ restricted to $W$. The three conditions above are the only ones you need to verify.

The subspace criterion

Conditions 2 and 3 together say that $W$ is closed under all linear combinations. This leads to a compact test:

Subspace criterion: A non-empty subset $W \subseteq V$ is a subspace of $V$ if and only if

$u, v \in W \text{ and } c, d \in F \implies cu + dv \in W.$

(Closed under all linear combinations.)

The condition also implies $\mathbf{0} \in W$ without needing to check it separately: since $W$ is non-empty, there exists some $u \in W$; taking $c = 0$ gives $0 \cdot u = \mathbf{0} \in W$. So you only need to verify that $W$ is non-empty and closed under linear combinations.

Examples

The trivial subspace

$W = \{\mathbf{0}\}$ is the smallest possible subspace of any vector space. It contains only the zero vector. It clearly satisfies all three conditions.

The whole space

$W = V$ itself is trivially a subspace — it is the largest.

Lines and planes through the origin in $\mathbb{R}^3$

• Any line through the origin in $\mathbb{R}^3$ — a set of the form $\{t\,v : t \in \mathbb{R}\}$ for a fixed nonzero $v$ — is a one-dimensional subspace.
• Any plane through the origin in $\mathbb{R}^3$ — a set of the form $\{s\,u + t\,v : s, t \in \mathbb{R}\}$ for linearly independent $u, v$ — is a two-dimensional subspace.

Notice the requirement “through the origin”: a line or plane that does not pass through the origin is not a subspace (it does not contain $\mathbf{0}$).

Solution sets of homogeneous systems

If $A \in M_{m,n}(F)$, the solution set of the homogeneous system $Ax = \mathbf{0}$ is a subspace of $F^n$. Check: $A\mathbf{0} = \mathbf{0}$ (zero is a solution); if $Ax = \mathbf{0}$ and $Ay = \mathbf{0}$ then $A(x+y) = \mathbf{0}$; if $Ax = \mathbf{0}$ then $A(cx) = c(Ax) = \mathbf{0}$. This subspace is the kernel of $A$, developed in Kernel.

Non-example

A line in $\mathbb{R}^2$ that does not pass through the origin, say $\{(x, y) : y = x + 1\}$, is not a subspace: it does not contain $(0, 0)$, and adding two points on it gives $(0, 0) + (1, 2) = (1, 2)$ — wait, more concretely: $(1, 2) + (2, 3) = (3, 5)$ lies on the line, but $2 \cdot (1, 2) = (2, 4)$ does not lie on $y = x + 1$. The closure conditions fail.

Spans and bases

From Linear Span, the span of any subset $S \subseteq V$ is automatically a subspace — the smallest subspace containing $S$. This gives a rich supply of subspaces from any set of vectors.

Bases of a subspace

A basis of a subspace $W$ is a subset $\mathcal{B} \subseteq W$ that is:

1. Linearly independent (as defined in Linearly Dependent).
2. Spanning: $\text{span}(\mathcal{B}) = W$.

A basis is a “minimal spanning set” and simultaneously a “maximal independent set” inside $W$. The number of elements in any basis of $W$ is always the same — this common number is the dimension of $W$, written $\dim W$.

Summary

• A subspace $W \subseteq V$ is a non-empty subset closed under addition and scalar multiplication; equivalently, closed under all linear combinations $cu + dv$.
• Being non-empty and closed under linear combinations automatically ensures $\mathbf{0} \in W$ and all vector space axioms.
• Key examples: $\{\mathbf{0}\}$, $V$ itself, lines/planes through the origin, and solution sets of homogeneous systems.
• A subset that does not contain $\mathbf{0}$ (such as a shifted affine subspace) is not a subspace.
• The span of any set $S$ (Linear Span) is the smallest subspace containing $S$.
• A basis of $W$ is a linearly independent spanning set; its size is $\dim W$.