K^n/Vector as linear combination of vectors/System of linear equations/Example

Let ${}K$ denote a field and ${}m\in \mathbb {N}$. Suppose that in ${}K^{m}$, there are ${}n$ vectors (or ${}m$-tuples)

$v_{1}={\begin{pmatrix}a_{11}\\a_{21}\\\vdots \\a_{m1}\end{pmatrix}},\,v_{2}={\begin{pmatrix}a_{12}\\a_{22}\\\vdots \\a_{m2}\end{pmatrix}},\ldots ,v_{n}={\begin{pmatrix}a_{1n}\\a_{2n}\\\vdots \\a_{mn}\end{pmatrix}}$

given. Let

${}w={\begin{pmatrix}c_{1}\\c_{2}\\\vdots \\c_{m}\end{pmatrix}}\,$

be another vector. We want to know whether ${}w$ can be written as a linear combination of the ${}v_{j}$. Thus, we are dealing with the question whether there are ${}n$ elements ${}s_{1},\ldots ,s_{n}\in K$, such that

${}s_{1}{\begin{pmatrix}a_{11}\\a_{21}\\\vdots \\a_{m1}\end{pmatrix}}+s_{2}{\begin{pmatrix}a_{12}\\a_{22}\\\vdots \\a_{m2}\end{pmatrix}}+\cdots +s_{n}{\begin{pmatrix}a_{1n}\\a_{2n}\\\vdots \\a_{mn}\end{pmatrix}}={\begin{pmatrix}c_{1}\\c_{2}\\\vdots \\c_{m}\end{pmatrix}}\,$

holds. This equality of vectors means identity in every component, so that this condition yields a system of linear equations

${\begin{matrix}a_{11}s_{1}+a_{12}s_{2}+\cdots +a_{1n}s_{n}&=&c_{1}\\a_{21}s_{1}+a_{22}s_{2}+\cdots +a_{2n}s_{n}&=&c_{2}\\\vdots &\vdots &\vdots \\a_{m1}s_{1}+a_{m2}s_{2}+\cdots +a_{mn}s_{n}&=&c_{m}.\end{matrix}}$