How do I prove that S is linearly independent?

Question

Let {$v_1$, $v_2$,..., $v_r$} be basis of V and {$w_1$, $w_2$,..., $w_m$} be basis of W. Denote S = {$v_1$, $v_2$,..., $v_r$, $w_1$, $w_2$,..., $w_m$}. Show that S is linearly independent.

Suppose I assume that V∩W = $\{0\},$ how do I go about showing that S is linearly independent, meaning $a_1v_1+a_2v_2+...+a_rv_r+b_1w_1+b_2w_2+...+b_mw_m = 0$?

In another case if I assume instead that dim(V∩W) ≥ 1, how do I show that S is also linearly independent?

Answer 1

The statement requires some adjustments, otherwise it makes no sense at all.

Let $V$ and $W$ be subspaces of the vector space $Z$ and suppose that $\{v_1,\dots,v_r\}$ is a basis of $V$ and $\{w_1,\dots,w_m\}$ is a basis of $W$. If $V\cap W=\{0\}$, then $\{v_1,\dots,v_r,w_1,\dots,w_m\}$ is linearly independent.

How to prove it? Suppose $a_1v_1+\dots+a_rv_r+b_1w_1+\dots+b_mw_m=0$. Set $$ z=a_1v_1+\dots+a_rv_r=-(b_1w_1+\dots+b_mw_m) $$ Then $z\in V\cap W$.

Can you finish?

By the way, the set $\{v_1,\dots,v_r,w_1,\dots,w_m\}$ is linearly independent if and only if $V\cap W=\{0\}$.