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Definition: Subspace

Let \((F,+,\cdot)\) be a field and \(V\) be a vector space over \(F\). A non-empty subset \(U\subseteq V\) is called subspace of \(V\), if

1. $0\in U,$
2. $x + y\in U$ for all $x,y\in U,$
3. $\alpha\cdot x\in U$ for all $\alpha \in F, x\in U.$

These properties are equivalent to those:

• \((U, + )\) is a subgroup of \((V, + )\) and
• \(U\) is closed under the scalar multiplication by elements of \(F\).

Table of Contents

Examples: 1

Mentioned in:

Definitions: 1 2 3 4
Lemmas: 5
Proofs: 6 7 8
Propositions: 9 10

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References

Bibliography

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