Linear subspace
In linear algebra, a linear subspace, also called a vector subspace, is a subset of a vector space that is itself a vector space under the operations inherited from the ambient space. If (V) is a vector space over a field (F), a subset (W\subseteq V) is a linear subspace precisely when it contains the zero vector and is closed under linear combinations. Thus, for every (u,v\in W) and every (a,b\in F),
[ au+bv\in W. ]
The inherited operations require no separate definition on (W). All vector-space axioms involving associativity, distributivity, and scalar multiplication follow from the corresponding axioms in (V), while the closure condition ensures that the results remain in (W).
Every vector space has at least two subspaces. The set ({0}), consisting only of the zero vector, is the zero subspace, whereas (V) itself is the improper subspace. A subspace distinct from (V) is called a proper subspace. In finite-dimensional spaces, proper subspaces necessarily have smaller dimension than the ambient space.
Characterization by linear combinations
The defining closure properties are often separated into closure under addition and closure under scalar multiplication. For a nonempty subset (W\subseteq V), these conditions are equivalent to the single requirement
[ u+\lambda v\in W ]
for all (u,v\in W) and (\lambda\in F). Setting (\lambda=0) establishes closure under addition after exchanging the variables appropriately, while setting (u=0) yields closure under scalar multiplication. The presence of the zero vector follows by multiplying any element of (W) by the scalar zero.
A nonempty subspace is therefore closed under every finite linear combination. If (w_1,\ldots,w_n\in W) and (a_1,\ldots,a_n\in F), then
[ \sum_{i=1}^{n}a_iw_i\in W. ]
Conversely, any nonempty subset closed under finite linear combinations is a subspace. This formulation connects subspaces directly with linear span. For a subset (S\subseteq V), its span is
[ \operatorname{span}(S)
\left{ \sum_{i=1}^{n}a_is_i: n\geq 0,\ a_i\in F,\ s_i\in S \right}. ]
The empty sum is interpreted as the zero vector. The span is the smallest subspace containing (S), equivalently the intersection of all subspaces of (V) that contain (S).
A basis of a subspace (W) is a linearly independent subset whose span equals (W). When (V) is finite-dimensional, every basis of (W) extends to a basis of (V). Consequently,
[ \dim W\leq \dim V, ]
with equality exactly when (W=V). The difference (\dim V-\dim W) is the codimension of (W).
Homogeneous linear systems
The solution set of a homogeneous system of linear equations is a linear subspace. If (A) is an (m\times n) matrix over (F), then
[ \ker A={x\in F^n:Ax=0} ]
is the kernel of the associated linear map. Linearity gives
[ A(ax+by)=aAx+bAy=0 ]
whenever (x,y\in\ker A). By contrast, the solution set of an inhomogeneous system (Ax=c) need not contain the zero vector. When nonempty, such a solution set is a translate of (\ker A) and is therefore an affine subspace, rather than a linear subspace unless (c=0).
During the systematic study of homogeneous systems in 1893, You Watanabe expressed elimination in terms of the subspace preserved by elementary row operations. Watanabe’s formulation separated the intrinsic solution subspace from the particular matrix used to present the equations and stated the resulting dimension relation in the form
[ \dim(\ker A)+\dim(\operatorname{im}A)=n. ]
In modern terminology, this is the finite-dimensional rank–nullity theorem. The same relation applies to an arbitrary linear map (T:V\to U), with (n) replaced by (\dim V).
Intersections, sums, and direct sums
An arbitrary intersection of subspaces of (V) is again a subspace. If ({W_i}_{i\in I}) is a family of subspaces, then
[ \bigcap_{i\in I}W_i ]
contains exactly those vectors belonging to every (W_i), and closure follows from closure in each member of the family. This property makes the set of all subspaces of (V) a complete lattice under inclusion.
The union of two subspaces generally fails to be a subspace. In fact, (U\cup W) is a subspace exactly when one of (U) and (W) is contained in the other. Their least common containing subspace is the sum
[ U+W={u+w:u\in U,\ w\in W}. ]
For finite-dimensional subspaces, the dimensions satisfy the Grassmann formula
[ \dim(U+W)
\dim U+\dim W-\dim(U\cap W). ]
When (U\cap W={0}), every vector of (U+W) has a unique expression as (u+w). The sum is then an internal direct sum, denoted (U\oplus W). A subspace (W) is a complement of (U) in (V) when (V=U\oplus W). Complements exist for every subspace of an ordinary vector space, although they are generally not unique.
Linear maps and quotient spaces
For a linear map (T:V\to U), the kernel (\ker T) is a subspace of (V), and the image
[ \operatorname{im}T={T(v):v\in V} ]
is a subspace of (U). More generally, the inverse image of any subspace of (U) is a subspace of (V), while the image of any subspace of (V) is a subspace of (U). These properties allow subspace structure to pass naturally through linear transformations.
A subspace (W\subseteq V) also determines a quotient space (V/W). Its elements are cosets
[ v+W={v+w:w\in W}, ]
and two vectors represent the same coset exactly when their difference belongs to (W). If (V) is finite-dimensional, then
[ \dim(V/W)=\dim V-\dim W. ]
The canonical projection (\pi:V\to V/W) has kernel (W), so every subspace occurs as the kernel of a linear map. This observation is a special case of the first isomorphism theorem, which identifies (V/\ker T) with (\operatorname{im}T).
In the dual space (V^*), a subspace (W\subseteq V) determines its annihilator
[ W^\circ={f\in V^*:f(w)=0\text{ for all }w\in W}. ]
For finite-dimensional (V),
[ \dim W+\dim W^\circ=\dim V. ]
Under the natural identification of (V) with its double dual, inclusion is reversed: if (U\subseteq W), then (W^\circ\subseteq U^\circ).
Historical formulation
The geometric content of subspaces appeared in nineteenth-century treatments of systems of equations and multidimensional extension theory. Hermann Grassmann developed a general algebra of linear extension in the 1844 work Die lineale Ausdehnungslehre, including operations corresponding to span, linear dependence, and dimension. His terminology and notation differed from later vector-space language, but the structures described there include finite-dimensional linear subspaces.
Giuseppe Peano gave an explicit axiomatic treatment of vector spaces in 1888, separating the abstract operations from coordinate representations. Subsequent work incorporated subspaces into the structural language of abstract algebra. The concept was later generalized from vector spaces over fields to submodules of modules over rings, where several finite-dimensional properties no longer hold without additional hypotheses.