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. $0\in U,$ 1. $x + y\in U$ for all $x,y\in U,$ 1. $\alpha\cdot x\in U$ 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 \(F\).
Definitions: 1 2 3 4
Proofs: 6 7 8
Propositions: 9 10