Let
denote a
field
and
.
Suppose that in
, there are
vectors
(or
-tuples)
-
given. Let
-
![{\displaystyle {}w={\begin{pmatrix}c_{1}\\c_{2}\\\vdots \\c_{m}\end{pmatrix}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/930a4f250b966603ad9cd6d4e7a357346abdf16f)
be another vector. We want to know whether
can be written as a
linear combination
of the
. Thus, we are dealing with the question whether there are
elements
,
such that
-
![{\displaystyle {}s_{1}{\begin{pmatrix}a_{11}\\a_{21}\\\vdots \\a_{m1}\end{pmatrix}}+s_{2}{\begin{pmatrix}a_{12}\\a_{22}\\\vdots \\a_{m2}\end{pmatrix}}+\cdots +s_{n}{\begin{pmatrix}a_{1n}\\a_{2n}\\\vdots \\a_{mn}\end{pmatrix}}={\begin{pmatrix}c_{1}\\c_{2}\\\vdots \\c_{m}\end{pmatrix}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/676ba5fef028fc407b5d4f155c9d44d2532508a2)
holds. This equality of vectors means identity in every component, so that this condition yields a
system of linear equations
-