Basis

Lemma

Let \(S\) be a Generating Set for \(W<V\) if for any \(T\subsetneq S\) \(T\) is not a generating set for \(W\), then \(S\) is Linear Independence

Suppose \(S\) was a linear dependent set.

Thus \(\exists v_i\in S\) and not zero \(a_i\in F\) st. \(\sum a_iv_i=0\)

As \(a_1\neq 0\), \(v_1=a_1^{-1}(-a_2v_2-...-a_nv_n)\)

Let \(T=S\backslash\{v_1\}\subsetneq S\)

Claim \(T\) is a generating set for \(V\)

This is a contradiction

A set \(B\subset V\) is called a basis if it is both a generating set of \(V\) and linearly independent set.