Subspaces

Precise definition

A subset $W$ of a vector space $V$ is a subspace if it contains zero and is closed under vector addition and scalar multiplication. Equivalently, for all $u,v\in W$ and scalars $a,b$, the combination $au+bv$ lies in $W$.

Notation and mathematical language

Kernels, ranges, spans, solution sets of homogeneous linear systems, and intersections of subspaces are subspaces. A solution set of $Ax=b$ with $b\ne0$ is generally affine rather than a subspace. The sum is $U+W=\{u+w\}$.

Conceptual picture

A subspace is a vector space using inherited operations. The combined linear-combination test is efficient because it verifies zero and both closure properties at once when the set is nonempty.

Conditions and key results

Equations defining a subspace must be homogeneous linear constraints. Nonlinear conditions such as $xy=0$, inequalities, fixed norms, and nonzero constants usually break closure. A union of two subspaces is a subspace only when one is contained in the other.

A reliable strategy

1. Identify the ambient space and check the subset is nonempty, usually by testing zero.
2. Take arbitrary members and scalars and apply the defining condition to their linear combination.
3. If it fails, give an explicit closure or zero counterexample; if it passes, find a basis by parameters or span.
4. Check dimension and distinguish the subspace from a translated affine set.

Fully worked example

Interpretation and application

Subspaces describe feasible directions, conserved constraints, data models, and invariant state sets. Affine feasible sets often arise in applications; translating by one known point converts them to a related homogeneous subspace.

Common mistakes

Verification and reasonableness

• Test arbitrary linear combinations symbolically.
• Use a basis description to verify every generated vector satisfies the defining constraints.
• Compare parameter count with rank–nullity.

Practice

1. Is $\{(x,y):x+y=0\}$ a subspace?
2. Is $\{(x,y):x+y=1\}$ a subspace?
3. Is an intersection of subspaces a subspace?

Further deduction

For finite-dimensional subspaces, $\dim(U+W)=\dim U+\dim W-\dim(U\cap W)$. A basis of the intersection is counted twice when dimensions are added, so it must be subtracted once. This formula is the subspace analogue of inclusion–exclusion and bounds how many genuinely new directions one subspace adds to another.

A direct sum $V=U\oplus W$ means every $v$ has a unique decomposition $u+w$. This requires $U+W=V$ and $U\cap W=\{0\}$. Dimension provides a shortcut in finite spaces: if the intersection is zero and dimensions add to $\dim V$, the sum is direct and covers all of $V$.

An affine solution set need not be a subspace. If $Ax=b$ is consistent, all solutions have form $x_p+\ker A$; this translate contains zero only when $b=0$ or zero happens to solve the equation. Thus homogeneous systems produce subspaces, while nonhomogeneous systems generally produce affine sets parallel to the null space. Testing closure on the translated set exposes why adding two particular solutions usually doubles the right-hand side instead of preserving it.

Related topics

• Vector Spaces
• Span
• Kernel and Range