Solution to Simultaneous Linear Equations

Theorem
Let $\displaystyle \forall i \in \left[{1 \,. \, . \, m}\right]: \sum _{j=1}^n {\alpha_{i j} x_j} = \beta_i$ 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.


 * $\displaystyle \sum_{j=1}^n {\alpha_{i j} x_j} = \beta_i$

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 $S$.