Editing Linear subspace (section)

==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 [[linear span|span]] of a collection of vectors, and the [[null space]], [[column space]], and [[row space]] of a [[matrix (mathematics)|matrix]]. Geometrically (especially over the field of real numbers and its subfields), a subspace is a [[flat (geometry)|flat]] in an ''n''-space that passes through the origin.

A natural description of a 1-subspace is the [[scalar multiplication]] of one non-[[additive identity|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:
:<math>\exist c\in K: \mathbf{v}' = c\mathbf{v}\text{ (or }\mathbf{v} = \frac{1}{c}\mathbf{v}'\text{)}</math>
This idea is generalized for higher dimensions with [[linear span]], but criteria for [[equality (mathematics)|equality]] of ''k''-spaces specified by sets of ''k'' vectors are not so simple.

A [[duality (mathematics)|dual]] description is provided with [[linear functionals]] (usually implemented as linear equations). One non-[[additive identity|zero]] linear functional '''F''' specifies its [[kernel (linear algebra)|kernel]] subspace '''F'''&nbsp;=&nbsp;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]]):
:<math>\exist c\in K: \mathbf{F}' = c\mathbf{F}\text{ (or }\mathbf{F} = \frac{1}{c}\mathbf{F}'\text{)}</math>
It is generalized for higher codimensions with a [[system of equations]]. The following two subsections will present this latter description in details, and [[#Span of vectors|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]] ''K''<sup>''n''</sup>:
<math display="block">\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\}. </math>

For example, the set of all vectors {{math|(''x'', ''y'', ''z'')}} (over real or [[rational number]]s) satisfying the equations
<math display="block">x + 3y + 2z = 0 \quad\text{and}\quad 2x - 4y + 5z = 0</math>
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''<sup>''k''</sup> will be the dimension of the [[null set]] of ''A'', the composite matrix of the ''n'' functions.

===Null space of a matrix===
{{main|Null space}}
In a finite-dimensional space, a homogeneous system of linear equations can be written as a single matrix equation:

:<math>A\mathbf{x} = \mathbf{0}.</math>

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

:<math>A = \begin{bmatrix} 1 & 3 & 2 \\ 2 & -4 & 5 \end{bmatrix} .</math>

Every subspace of ''K''<sup>''n''</sup> can be described as the null space of some matrix (see {{slink||Algorithms}} below for more).

===Linear parametric equations===
The subset of ''K''<sup>''n''</sup> described by a system of homogeneous linear [[parametric equations]] is a subspace:

:<math>\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\}. </math>

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

:<math>x = 2t_1 + 3t_2,\;\;\;\;y = 5t_1 - 4t_2,\;\;\;\;\text{and}\;\;\;\;z = -t_1 + 2t_2</math>

is a two-dimensional subspace of ''K''<sup>3</sup>, if ''K'' is a [[number field]] (such as real or rational numbers).<ref name="fields" group="note">Generally, ''K'' can be any field of such [[characteristic (algebra)|characteristic]] that the given integer matrix has the appropriate [[rank (matrix theory)|rank]] in it. All fields include [[integer]]s, but some integers may equal to zero in some fields.</ref>

===Span of vectors===
{{main|Linear span}}
In linear algebra, the system of parametric equations can be written as a single vector equation:

:<math>\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}.</math>

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

In general, a '''linear combination''' of vectors '''v'''<sub>1</sub>,&nbsp;'''v'''<sub>2</sub>,&nbsp;...&nbsp;,&nbsp;'''v'''<sub>''k''</sub> is any vector of the form

:<math>t_1 \mathbf{v}_1 + \cdots + t_k \mathbf{v}_k.</math>

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

:<math>\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\} .</math>

If the vectors '''v'''<sub>1</sub>,&nbsp;...&nbsp;,&nbsp;'''v'''<sub>''k''</sub> have ''n'' components, then their span is a subspace of ''K''<sup>''n''</sup>. Geometrically, the span is the flat through the origin in ''n''-dimensional space determined by the points '''v'''<sub>1</sub>,&nbsp;...&nbsp;,&nbsp;'''v'''<sub>''k''</sub>.

; Example
: The ''xz''-plane in '''R'''<sup>3</sup> can be parameterized by the equations
::<math>x = t_1, \;\;\; y = 0, \;\;\; z = t_2.</math>

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

::<math>(t_1, 0, t_2) = t_1(1,0,0) + t_2(0,0,1)\text{.}</math>

: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,&nbsp;0,&nbsp;0) and then moving some distance in the direction of (0,&nbsp;0,&nbsp;1).

===Column space and row space===
{{main|Row and column spaces}}
A system of linear parametric equations in a finite-dimensional space can also be written as a single matrix equation:

:<math>\mathbf{x} = A\mathbf{t}\;\;\;\;\text{where}\;\;\;\;A = \left[ \begin{alignat}{2} 2 && 3 & \\ 5 && \;\;-4 & \\ -1 && 2 & \end{alignat} \,\right]\text{.}</math>

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 (mathematics)|image]]) of the matrix ''A''. It is precisely the subspace of ''K''<sup>''n''</sup> 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).

===Independence, basis, and dimension===
{{main|Linear independence|Basis (linear algebra)|Dimension (vector space)}}
[[File:Basis for a plane.svg|thumb|280px|right|The vectors '''u''' and '''v''' are a basis for this two-dimensional subspace of '''R'''<sup>3</sup>.]]
In general, a subspace of ''K''<sup>''n''</sup> 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''<sup>3</sup> spanned by the three vectors (1,&nbsp;0,&nbsp;0), (0,&nbsp;0,&nbsp;1), and (2,&nbsp;0,&nbsp;3) is just the ''xz''-plane, with each point on the plane described by infinitely many different values of {{nowrap| ''t''<sub>1</sub>, ''t''<sub>2</sub>, ''t''<sub>3</sub>}}.

In general, vectors '''v'''<sub>1</sub>,&nbsp;...&nbsp;,&nbsp;'''v'''<sub>''k''</sub> are called '''linearly independent''' if

:<math>t_1 \mathbf{v}_1 + \cdots + t_k \mathbf{v}_k \;\ne\; u_1 \mathbf{v}_1 + \cdots + u_k \mathbf{v}_k</math>

for
(''t''<sub>1</sub>,&nbsp;''t''<sub>2</sub>,&nbsp;...&nbsp;,&nbsp;''t<sub>k</sub>'')&nbsp;≠ (''u''<sub>1</sub>,&nbsp;''u''<sub>2</sub>,&nbsp;...&nbsp;,&nbsp;''u<sub>k</sub>'').<ref group="note">This definition is often stated differently: vectors '''v'''<sub>1</sub>, ..., '''v'''<sub>''k''</sub> are linearly independent if
{{nowrap| ''t''<sub>1</sub>'''v'''<sub>1</sub> + ··· + ''t<sub>k</sub>'''''v'''<sub>''k''</sub> ≠ '''0'''}} for {{nowrap| (''t''<sub>1</sub>, ''t''<sub>2</sub>, ..., ''t<sub>k</sub>'') ≠ (0, 0, ..., 0)}}. The two definitions are equivalent.</ref>
If {{nowrap| '''v'''<sub>1</sub>, ..., '''v'''<sub>''k''</sub> }} are linearly independent, then the '''coordinates''' {{nowrap| ''t''<sub>1</sub>, ..., ''t<sub>k</sub>''}} 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|§ Algorithms]] below for more).

; Example
: Let ''S'' be the subspace of '''R'''<sup>4</sup> defined by the equations
::<math>x_1 = 2 x_2\;\;\;\;\text{and}\;\;\;\;x_3 = 5x_4.</math>
:Then the vectors (2,&nbsp;1,&nbsp;0,&nbsp;0) and (0,&nbsp;0,&nbsp;5,&nbsp;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:

::<math>(2t_1, t_1, 5t_2, t_2) = t_1(2, 1, 0, 0) + t_2(0, 0, 5, 1).</math>

:The subspace ''S'' is two-dimensional. Geometrically, it is the plane in '''R'''<sup>4</sup> passing through the points (0,&nbsp;0,&nbsp;0,&nbsp;0), (2,&nbsp;1,&nbsp;0,&nbsp;0), and (0,&nbsp;0,&nbsp;5,&nbsp;1).