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Solvability of a LSE

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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$).
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  automatisch erstellt am 19.  8. 2013