# Help proving a set of vectors is a span of vector space :

I know that $S(v_1,v_2,v_3)$ is the span of the vector space $V$.

Now I need to show that $S'(v_1,v_2,v_1+v_3)$ is also the span of vector space $V$.

**It's actually prove or disprove but I'm pretty sure it's true so I need to prove.

I'm not sure what to do, how do I prove it's a span of $V$ ? If the first statement only means that $a_1\cdot v_1 + a_2\cdot v_2 + a_3\cdot v_3$ is all linear combinations of $V$.

If I look at all linear combinations of $S'$, I see $(4b_1+b_3)\cdot v_1 + b_\cdot v_2 + b_3\cdot v_3$. Now I see it looks the same, I don't know if it's a proof or not, but it looks the same to me, scalar multiplied by the vector, how do I put the proof if it's a proof to words? I'm not sure what I need to do in this exercise.

Probably you mean V is the span of $\{v_1, v_2,v_3\}$.
This is so because $v_1+v_3\in S(v_1, v_2,v_3)$. Conversely $v_3\in S(v_1, v_2,v_1+v_3)$ since we can write $$v_3=(v_1+v_3)-v_1.$$ Thus the generators of each span belong to the other span. Hence they're equal.