Linear subspace

Then W is a subspace of V. Proof: Let the field be R again, but now let the vector space V be the Cartesian plane R2.

Proof: In general, any subset of the real coordinate space Rn that is defined by a homogeneous system of linear equations will yield a subspace.

Proof: Keep the same field and vector space as before, but now consider the set Diff(R) of all differentiable functions.

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.

[8] 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.

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.

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): It is generalized for higher codimensions with a system of equations.

The solution set to any homogeneous system of linear equations with n variables is a subspace in the coordinate space Kn:

For example, the set of all vectors (x, y, z) (over real or rational numbers) satisfying the equations

For example, the subspace described above is the null space of the matrix Every subspace of Kn can be described as the null space of some matrix (see § Algorithms below for more).

The subset of Kn described by a system of homogeneous linear parametric equations is a subspace: For example, the set of all vectors (x, y, z) parameterized by the equations is a two-dimensional subspace of K3, if K is a number field (such as real or rational numbers).

[note 2] In linear algebra, the system of parametric equations can be written as a single vector equation: The expression on the right is called a linear combination of the vectors (2, 5, −1) and (3, −4, 2).

Geometrically, the span is the flat through the origin in n-dimensional space determined by the points v1, ... , vk.

A system of linear parametric equations in a finite-dimensional space can also be written as a single matrix equation: 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 Kn spanned by the column vectors of A.

In general, a subspace of Kn determined by k parameters (or spanned by k vectors) has dimension k. However, there are exceptions to this rule.

For example, the subspace of K3 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, ... , vk are called linearly independent if for (t1, t2, ... , tk) ≠ (u1, u2, ... , uk).

[note 3] If v1, ..., vk are linearly independent, then the coordinates t1, ..., tk for a vector in the span are uniquely determined.

Any spanning set for a subspace can be changed into a basis by removing redundant vectors (see § Algorithms below for more).

The set-theoretical inclusion binary relation specifies a partial order on the set of all subspaces (of any dimension).

Here, the minimum only occurs if one subspace is contained in the other, while the maximum is the most general case.

[23] Applying orthogonal complements twice returns the original subspace:

[citation needed] In spaces with other bilinear forms, some but not all of these results still hold.

[citation needed] Most algorithms for dealing with subspaces involve row reduction.

If the final column of the reduced row echelon form contains a pivot, then the input vector v does not lie in S. See the article on null space for an example.