Proposition (Criterion for a basis)

A set of vectors $\{v_1,...,v_n\}$ in a Vector Space V is a basis if every element $v\in V$ can be be uniquely written as a linear combination $$a_1v_1+...+a_nv_n$$

Proposition (All bases have the same size)

Any two bases of $V$ have the same number of elements.

Proposition (Every spanning set contains a basis)

If $V= span(v_1,...,v_k)$, then the set of vectors $B=\{v_1,...,v_k\}$ can be reduced to a Finite Basis.

Proposition (Linearly independent set can be extended to a basis)

If $V$ is a finite dimensional Vector Space, then every set of linearly independent vectors in V can be extended to a Finite Basis.