Linear subspace

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.

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.

In the vector space V = R (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:

1. 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.
2. 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.

Let the field be R again, but now let the vector space V be the Cartesian plane R. Take W to be the set of points (x, y) of R such that x = y. Then W is a subspace of R.

Proof:

1. 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.
2. 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 R 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.)

Again take the field to be R, but now let the vector space V be the set R of all functions from R to R. Let C(R) be the subset consisting of continuous functions. Then C(R) is a subspace of R.

Proof:

1. We know from calculus that 0 ∈ C(R) ⊂ R.
2. We know from calculus that the sum of continuous functions is continuous.
3. Again, we know from calculus that the product of a continuous function and a number is continuous.

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.

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 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:

\(\exist c\in K: \mathbf{v}' = c\mathbf{v}\text{ (or }\mathbf{v} = \frac{1}{c}\mathbf{v}'\text{)}\)

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):

\(\exist c\in K: \mathbf{F}' = c\mathbf{F}\text{ (or }\mathbf{F} = \frac{1}{c}\mathbf{F}'\text{)}\)

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.

The solution set to any homogeneous system of linear equations with n variables is a subspace in the coordinate space K: \[\left\{ \left[\!\! \begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array} \!\!\right] \in K^n : \begin{alignat}{6} a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \cdots + \;&& a_{1n} x_n &&\; = 0& \\ a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \cdots + \;&& a_{2n} x_n &&\; = 0& \\ && && && && && \vdots\quad& \\ a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \cdots + \;&& a_{mn} x_n &&\; = 0& \end{alignat} \right\}.\]

For example, the set of all vectors (x, y, z) (over real or rational numbers) satisfying the equations \[x + 3y + 2z = 0 \quad\text{and}\quad 2x - 4y + 5z = 0\] is a one-dimensional subspace. More generally, that is to say that given a set of n independent functions, the dimension of the subspace in K will be the dimension of the null set of A, the composite matrix of the n functions.

In a finite-dimensional space, a homogeneous system of linear equations can be written as a single matrix equation:

\(A\mathbf{x} = \mathbf{0}.\)

The set of solutions to this equation is known as the null space of the matrix. For example, the subspace described above is the null space of the matrix

\(A = \begin{bmatrix} 1 & 3 & 2 \\ 2 & -4 & 5 \end{bmatrix} .\)

Every subspace of K can be described as the null space of some matrix (see § Algorithms below for more).

The subset of K described by a system of homogeneous linear parametric equations is a subspace:

\(\left\{ \left[\!\! \begin{array}{c} x_1 \\ x_2 \\ \vdots \\ x_n \end{array} \!\!\right] \in K^n : \begin{alignat}{7} x_1 &&\; = \;&& a_{11} t_1 &&\; + \;&& a_{12} t_2 &&\; + \cdots + \;&& a_{1m} t_m & \\ x_2 &&\; = \;&& a_{21} t_1 &&\; + \;&& a_{22} t_2 &&\; + \cdots + \;&& a_{2m} t_m & \\ && \vdots\;\; && && && && && & \\ x_n &&\; = \;&& a_{n1} t_1 &&\; + \;&& a_{n2} t_2 &&\; + \cdots + \;&& a_{nm} t_m & \\ \end{alignat} \text{ for some } t_1,\ldots,t_m\in K \right\}.\)

For example, the set of all vectors (x, y, z) parameterized by the equations

\(x = 2t_1 + 3t_2,\;\;\;\;y = 5t_1 - 4t_2,\;\;\;\;\text{and}\;\;\;\;z = -t_1 + 2t_2\)

is a two-dimensional subspace of K, if K is a number field (such as real or rational numbers).

In linear algebra, the system of parametric equations can be written as a single vector equation:

\(\begin{bmatrix} x \\ y \\ z \end{bmatrix} \;=\; t_1 \!\begin{bmatrix} 2 \\ 5 \\ -1 \end{bmatrix} + t_2 \!\begin{bmatrix} 3 \\ -4 \\ 2 \end{bmatrix}.\)

The expression on the right is called a linear combination of the vectors (2, 5, −1) and (3, −4, 2). These two vectors are said to span the resulting subspace.

In general, a linear combination of vectors v₁, v₂, ... , v_k is any vector of the form

\(t_1 \mathbf{v}_1 + \cdots + t_k \mathbf{v}_k.\)

The set of all possible linear combinations is called the span:

\(\text{Span} \{ \mathbf{v}_1, \ldots, \mathbf{v}_k \} = \left\{ t_1 \mathbf{v}_1 + \cdots + t_k \mathbf{v}_k : t_1,\ldots,t_k\in K \right\} .\)

If the vectors v₁, ... , v_k have n components, then their span is a subspace of K. Geometrically, the span is the flat through the origin in n-dimensional space determined by the points v₁, ... , v_k.

Example

The xz-plane in R can be parameterized by the equations

\(x = t_1, \;\;\; y = 0, \;\;\; z = t_2.\)

As a subspace, the xz-plane is spanned by the vectors (1, 0, 0) and (0, 0, 1). Every vector in the xz-plane can be written as a linear combination of these two:

\((t_1, 0, t_2) = t_1(1,0,0) + t_2(0,0,1)\text{.}\)

Geometrically, this corresponds to the fact that every point on the xz-plane can be reached from the origin by first moving some distance in the direction of (1, 0, 0) and then moving some distance in the direction of (0, 0, 1).

A system of linear parametric equations in a finite-dimensional space can also be written as a single matrix equation:

\(\mathbf{x} = A\mathbf{t}\;\;\;\;\text{where}\;\;\;\;A = \left[ \begin{alignat}{2} 2 && 3 & \\ 5 && \;\;-4 & \\ -1 && 2 & \end{alignat} \,\right]\text{.}\)

In this case, the subspace consists of all possible values of the vector x. In linear algebra, this subspace is known as the column space (or image) of the matrix A. It is precisely the subspace of K spanned by the column vectors of A.

The row space of a matrix is the subspace spanned by its row vectors. The row space is interesting because it is the orthogonal complement of the null space (see below).

In general, a subspace of K determined by k parameters (or spanned by k vectors) has dimension k. However, there are exceptions to this rule. For example, the subspace of K spanned by the three vectors (1, 0, 0), (0, 0, 1), and (2, 0, 3) is just the xz-plane, with each point on the plane described by infinitely many different values of t₁, t₂, t₃.

In general, vectors v₁, ... , v_k are called linearly independent if

\(t_1 \mathbf{v}_1 + \cdots + t_k \mathbf{v}_k \;\ne\; u_1 \mathbf{v}_1 + \cdots + u_k \mathbf{v}_k\)

for (t₁, t₂, ... , t_k) ≠ (u₁, u₂, ... , u_k). If v₁, ..., v_k are linearly independent, then the coordinates t₁, ..., t_k for a vector in the span are uniquely determined.

A basis for a subspace S is a set of linearly independent vectors whose span is S. The number of elements in a basis is always equal to the geometric dimension of the subspace. Any spanning set for a subspace can be changed into a basis by removing redundant vectors (see § Algorithms below for more).

Example

Let S be the subspace of R defined by the equations

\(x_1 = 2 x_2\;\;\;\;\text{and}\;\;\;\;x_3 = 5x_4.\)

Then the vectors (2, 1, 0, 0) and (0, 0, 5, 1) are a basis for S. In particular, every vector that satisfies the above equations can be written uniquely as a linear combination of the two basis vectors:

\((2t_1, t_1, 5t_2, t_2) = t_1(2, 1, 0, 0) + t_2(0, 0, 5, 1).\)

The subspace S is two-dimensional. Geometrically, it is the plane in R passing through the points (0, 0, 0, 0), (2, 1, 0, 0), and (0, 0, 5, 1).