Editing Linear subspace (section)

===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).