Linear and multilinear algebra; matrix theory Definition

Homogeneous system of linear equations

Definition

Let $F$ be a field. A system of linear equations over $F$ is homogeneous when each of its constant terms is $0$ . In matrix form, a homogeneous system is an equation $Ax = 0$ , where $A$ is an $m \times n$ matrix over $F$ and $x = (x_1, \dots, x_n)^T$ .

Remarks

The tuple $(0, \dots, 0)$ is always a solution, the trivial solution. The solution set of $Ax = 0$ is a linear subspace of $F^n$ , the null space of $A$ . If a system $Ax = b$ has a solution $p$ , then its solution set is $\{p + v : Av = 0\}$ . A homogeneous system with more unknowns than equations has a nontrivial solution.

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Linear equation definition
System of linear equations definition
Homogeneous system of linear equations definition

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Tag	`06JT7KDHXM`
URI	`m2://06JT7KDHXM`
Kind	definition
Writing	W0 stub
MSC	15A06
Defines	Homogeneous system of linear equations
Uses	System of linear equations

Homogeneous system of linear equations

Definition

Remarks

Reading path

Where this leads

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