Solution to Simultaneous Linear Equations
Theorem
Let $\ds \forall i \in \closedint 1 m: \sum _{j \mathop = 1}^n {\alpha_{i j} x_j} = \beta_i$ 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 $x = \tuple {x_1, x_2, \ldots, x_n}$ is a solution of this system if and only if:
- $\sqbrk \alpha_{m n} \sqbrk x_{n 1} = \sqbrk \beta_{m 1}$
where $\sqbrk a_{m n}$ is an $m \times n$ matrix.
Proof
We can see the truth of this by writing them out in full.
- $\ds \sum_{j \mathop = 1}^n {\alpha_{i j} x_j} = \beta_i$
can be written as:
| \(\ds \alpha_{1 1} x_1 + \alpha_{1 2} x_2 + \ldots + \alpha_{1 n} x_n\) | \(=\) | \(\ds \beta_1\) | ||||||||||||
| \(\ds \alpha_{2 1} x_1 + \alpha_{2 2} x_2 + \ldots + \alpha_{2 n} x_n\) | \(=\) | \(\ds \beta_2\) | ||||||||||||
| \(\ds \) | \(\vdots\) | \(\ds \) | ||||||||||||
| \(\ds \alpha_{m 1} x_1 + \alpha_{m 2} x_2 + \ldots + \alpha_{m n} x_n\) | \(=\) | \(\ds \beta_m\) |
while $\sqbrk \alpha_{m n} \sqbrk x_{n 1} = \sqbrk \beta_{m 1}$ can be written as:
- $\begin {bmatrix}
\alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} \\ \alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
\alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} \end {bmatrix} \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end {bmatrix} = \begin {bmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end {bmatrix}$
So the question:
- Find a solution to the following system of $m$ simultaneous linear equations in $n$ variables
is equivalent to:
- Given the following element $\mathbf A \in \map {\MM_K} {m, n}$ and $\mathbf b \in \map {\MM_K} {m, 1}$, find the set of all $\mathbf x \in \map {\MM_K} {n, 1}$ such that $\mathbf A \mathbf x = \mathbf b$
where $\map {\MM_K} {m, n}$ is the $m \times n$ matrix space over $S$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous): Chapter $\text {V}$: Vector Spaces: $\S 30$. Linear Equations