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Mathematics-Online course: Preparatory Course Mathematics - Linear Algebra and Geometry - Systems of Linear Equations

Solvability of a linear system of equations


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The solution set of a homogeneous LSE

$\displaystyle Ax = 0
$

with $ m\times n$-coefficient matrix $ A$ forms a subspace $ U$ of $ K^n$.

If the inhomogeneous LSE

$\displaystyle Ax = b
$

has a solution $ v$, then for the general solution we have

$\displaystyle x \in v + U\,
,
$

that is, the solution set is an affine subspace of $ K^n$. In particular, an inhomogeneous LSE can have either no solution, or exactly one solution ($ U=\{0\}$), or an infinite number of solutions ( $ \operatorname{dim} U>0$).
(Authors: App/Burkhardt/Höllig/Kimmerle)

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  automatically generated 1/9/2017