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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \alpha_{11} x_1 + \alpha_{12} x_2 + \ldots + \alpha_{1n} x_n\)	\(=\)	\(\displaystyle \beta_1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \alpha_{21} x_1 + \alpha_{22} x_2 + \ldots + \alpha_{2n} x_n\)	\(=\)	\(\displaystyle \beta_2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\vdots\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \alpha_{m1} x_1 + \alpha_{m2} x_2 + \ldots + \alpha_{mn} x_n\)	\(=\)	\(\displaystyle \beta_m\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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 $S$.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965): $\S 30$