Question

Good morning.

My question is:

Suppose v₁; v₂;...; v_m is linearly independent in V and w ∈ V .

Show that v₁; v₂; ...; v_m; w is linearly independent if and only if w ∉ span(v₁; v₂; :::; v_m).

Please assist.

Answer 1

Proof:

First suppose $v_1,v_2,....,v_m$ is linearly independent. Then if w$\in span(v_1,...,v_m),$ we can write w as linear combination of $v_1,v_2,....,v_m$ that is $w=a_1v_1+...+a_mv_m$

Adding both sides of the equation by -w ,we have

$$a_1v_1+....+a_mv_m+(-w)=0$$

Therefore we can write zero as $a_1v_1+....+a_mv_m+(-w)$ , so there exists $a_1,a_2,...a_m,-1,$

not all 0, such that $a_1v_1+....+a_mv_m+(-w)=0$ . by the definition of linear dependence, we have $v_1,...,v_m,w$ is linearly dependent, which contradicts our initial assumption. Thus we have $w\notin span(v_1,v_2,....,v_m)$