Solution to Simultaneous Linear Equations

Theorem
Let $$\forall i \in \left[{1 \,. \, . \, m}\right]: \sum \limits_{j=1}^n {\alpha_{i j} x_j} = \beta_i$$ be a system of simultaneous linear equations.

where all of $$\alpha_1, \ldots, a_n, x_1, \ldots x_n, \beta_i, \ldots, \beta_m$$ are elements of a field $$K$$.

Then $$x = \left({x_1, x_2, \ldots, x_n}\right)$$ is a solution of this system iff:


 * $$\left[{\alpha}\right]_{m n} \left[{x}\right]_{n 1} = \left[{\beta}\right]_{m 1}$$

where $$\left[{a}\right]_{m n}$$ is an $m \times n$ matrix.

Proof
We can see the truth of this by writing them out in full.


 * $$\sum \limits_{j=1}^n {\alpha_{i j} x_j} = \beta_i$$

can be written as:

$$ $$ $$ $$

while $$\left[{\alpha}\right]_{m n} \left[{x}\right]_{n 1} = \left[{\beta}\right]_{m 1}$$ can be written as:


 * $$\begin{bmatrix}

\alpha_{11} & \alpha_{12} & \cdots & \alpha_{1n} \\ \alpha_{21} & \alpha_{22} & \cdots & \alpha_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{m1} & \alpha_{m2} & \cdots & \alpha_{mn} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end{bmatrix} $$

So the question:
 * Find a solution to the following system of $$m$$ simultaneous equations in $$n$$ variables

is equivalent to:
 * Given the following element $$\mathbf A \in \mathcal M_K \left({m, n}\right)$$ and $$\mathbf b \in \mathcal M_K \left({m, 1}\right)$$, find the set of all $$\mathbf x \in \mathcal M_K \left({n, 1}\right)$$ such that $$\mathbf A \mathbf x = \mathbf b$$

where $$\mathcal M_K \left({m, n}\right)$$ is the $m \times n$ matrix space over $$S$$.