Linear Independence

Given a vector space $V$ and a finite set $S=\{v_1,v_2,\dots ,v_n\}$, $S$ is linear independent if the only solution to $\sum _{i=1}^n{c}_i{v}_1=0$ occurs when $c_1=c_2=\cdots =c_n=0$

Thm: Unique Linear Combinations

If $V$ is a vector space and $S=\{v_1,\dots v_n\}$ then $S$ is linearly independent if and only if $\mathrm{\forall }w\in \text{span}(S)$ there is a unique set of $c_i,i=1,\dots ,n$ such that $w=\sum _i^n{c}_i{v}_i$

Thm: Fundamental Theorem

if $V$ is a vector space and $S$ is a spanning set and $T$ is linearly independent then $|T|\le |S|$
by Exchange Theorem

Thm: Exchange Theorem

If $w\in \text{span}\{v_1,\dots ,v_n\}$ then $\mathrm{\exists }i\in \{1,\dots ,n\}$ s.t $\text{span}\{v_1,\dots ,v_n\}$ = $\text{span}\{v_1,\dots ,v_{i-1},w,v_{i+1},\dots ,v_n\}$