Definition: Generating Systems

Let \(A\subset V\) be a subset of a vector space \(V\) over a field \(F\) and let \(U\subset V\) be a subspace of \(V\). Then \(A\) is called the generating system of \(U\) if and only its linear span equals \(U\): \[A\text{ is generating system of }U\Longleftrightarrow \operatorname{Span}(A)=U.\]

This is equivalent to the property that each vector \(x\in U\) can be represented as a linear combination of a given finitely many vectors in \(v_i\in A\), i.e.

\[x=\sum _{j\in J}s_{j}v_{j}\]

for all \(x\in U\) with \(J\) being a finite index set, all \(v_i\in A\) and with \(s_{j}\in F\).

If \(A\) is a generating system of \(U\) (like defined above), the subspace \(U\) is said to be generated by the vectors in \(A\).

Definitions: 1 2
Proofs: 3 4

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  1. Koecher Max: "Lineare Algebra und analytische Geometrie", Springer-Verlag Berlin Heidelberg New York, 1992, 3rd Volume