True or false (linear dependence)?

Question

Is this sentence true or false ? if true please prove that.

$V=\{v_1,v_2,v_3,\ldots,v_k\}$ is linearly independent and $w_1,w_2\notin \operatorname{span} \{V\}$

then $\{w_1-w_2,v_1,v_2,\ldots,v_k\}$ is linearly independent.

Answer 1

False. Consider $V= \{(1,0,0),(0,1,0) \}$ which is linearly independent.

Now consider $w_1= (2,0,1),$ $w_2 = (0,0,1)$. This satisfies that $w_1$and $w_2 $ are not in the span of $V$.

But $w_1-w_2=(2,0,0)=2(1,0,0)$, hence adding $w_1-w_2$ to $V$ would make it a linearly dependent set.

Answer 2

HINT

This is not true in general, let consider the case

$$w_1=w_2$$