In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.
Definition If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations of V. Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w1, w2 are elements of W and α, β are elements of K, it follows that αw1 + βw2 is in W. The singleton set consisting of the zero vector alone and the entire vector space itself are linear subspaces that are called the trivial subspaces of the vector space.
Examples
Example I In the vector space V = R3 (the real coordinate space over the field R of real numbers), take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V. Proof:
Given u and v in W, then they can be expressed as u = (u1, u2, 0) and v = (v1, v2, 0). Then u + v = (u1+v1, u2+v2, 0+0) = (u1+v1, u2+v2, 0). Thus, u + v is an element of W, too. Given u in W and a scalar c in R, if u = (u1, u2, 0) again, then cu = (cu1, cu2, c0) = (cu1, cu2,0). Thus, cu is an element of W too.
Example II Let the field be R again, but now let the vector space V be the Cartesian plane R2. Take W to be the set of points (x, y) of R2 such that x = y. Then W is a subspace of R2. Proof:
Let p = (p1, p2) and q = (q1, q2) be elements of W, that is, points in the plane such that p1 = p2 and q1 = q2. Then p + q = (p1+q1, p2+q2); since p1 = p2 and q1 = q2, then p1 + q1 = p2 + q2, so p + q is an element of W. Let p = (p1, p2) be an element of W, that is, a point in the plane such that p1 = p2, and let c be a scalar in R. Then cp = (cp1, cp2); since p1 = p2, then cp1 = cp2, so cp is an element of W. In general, any subset of the real coordinate space Rn that is defined by a homogeneous system of linear equations will yield a subspace. (The equation in example I was z = 0, and the equation in example II was x = y.)
Example III Again take the field to be R, but now let the vector space V be the set RR of all functions from R to R. Let C(R) be the subset consisting of continuous functions. Then C(R) is a subspace of RR. Proof:
We know from calculus that 0 ∈ C(R) ⊂ RR. We know from calculus that the sum of continuous functions is continuous. Again, we know from calculus that the product of a continuous function and a number is continuous.
Example IV Keep the same field and vector space as before, but now consider the set Diff(R) of all differentiable functions. The same sort of argument as before shows that this is a subspace too. Examples that extend these themes are common in functional analysis.
Properties of subspaces From the definition of vector spaces, it follows that subspaces are nonempty, and are closed under sums and under scalar multiples. Equivalently, subspaces can be characterized by the property of being closed under linear combinations. That is, a nonempty set W is a subspace if and only if every linear combination of finitely many elements of W also belongs to W. The equivalent definition states that it is also equivalent to consider linear combinations of two elements at a time. In a topological vector space X, a subspace W need not be topologically closed, but a finite-dimensional subspace is always closed. The same is true for subspaces of finite codimension (i.e., subspaces determined by a finite number of continuous linear functionals).
Descriptions Descriptions of subspaces include the solution set to a homogeneous system of linear equations, the subset of Euclidean space described by a system of homogeneous linear parametric equations, the span of a collection of vectors, and the null space, column space, and row space of a matrix. Geometrically (especially over the field of real numbers and its subfields), a subspace is a flat in an n-space that passes through the origin. A natural description of a 1-subspace is the scalar multiplication of one non-zero vector v to all possible scalar values. 1-subspaces specified by two vectors are equal if and only if one vector can be obtained from another with scalar multiplication:
∃ c ∈ K : v ′ = c v (or v = 1 c v ′ ) {\displaystyle \exists c\in K:\mathbf {v} '=c\mathbf {v} {\text{ (or }}\mathbf {v} ={\frac {1}{c}}\mathbf {v} '{\text{)}}}
This idea is generalized for higher dimensions with linear span, but criteria for equality of k-spaces specified by sets of k vectors are not so simple. A dual description is provided with linear functionals (usually implemented as linear equations). One non-zero linear functional F specifies its kernel subspace F = 0 of codimension 1. Subspaces of codimension 1 specified by two linear functionals are equal, if and only if one functional can be obtained from another with scalar multiplication (in the dual space):
∃ c ∈ K : F ′ = c F (or F = 1 c F ′ ) {\displaystyle \exists c\in K:\mathbf {F} '=c\mathbf {F} {\text{ (or }}\mathbf {F} ={\frac {1}{c}}\mathbf {F} '{\text{)}}}
It is generalized for higher codimensions with a system of equations. The following two subsections will present this latter description in details, and the remaining four subsections further describe the idea of linear span.
Systems of linear equations The solution set to any homogeneous system of linear equations with n variables is a subspace in the coordinate space Kn:
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