Linear and multilinear algebra; matrix theory Definition

System of linear equations

Definition

Let $F$ be a field and let $m, n \ge 1$ be integers. Consider a system of linear equations over $F$ , with $m$ equations in the unknowns $x_1, \dots, x_n$ . Such a system is a list of linear equations $\begin{aligned} a_{11} x_1 + a_{12} x_2 + \cdots + a_{1n} x_n &= b_1, \\ &\ \ \vdots \\ a_{m1} x_1 + a_{m2} x_2 + \cdots + a_{mn} x_n &= b_m, \end{aligned}$ where all $a_{ij}$ and $b_i$ are elements of $F$ . A solution of the system is a tuple $(s_1, \dots, s_n) \in F^n$ that is a solution of each of the $m$ equations. The solution set of the system is the set of all its solutions.

Remarks

With the coefficient matrix $A = (a_{ij})$ , the vector $b = (b_1, \dots, b_m)^T$ and $x = (x_1, \dots, x_n)^T$ , the system is the single equation $Ax = b$ . Over a field, a system has no solution, exactly one solution, or more than one solution. When $F$ is infinite, "more than one" means infinitely many.

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Linear equation definition
System of linear equations definition

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Tag	`0JTRMBQ1SA`
URI	`m2://0JTRMBQ1SA`
Kind	definition
Writing	W1 stated
MSC	15A06
Defines	System of linear equations
Uses	Linear equation

System of linear equations

Definition

Remarks

Reading path

Where this leads

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