Are the vector subspaces equal?

Question

Let $v_1, v_2, v_3, v_4$ be elements of a vector space.

Does it hold that $\langle 2v_1, 3v_2, \frac{1}{2}v_2+v_3, -3v_4\rangle=\langle v_1, v_2, v_3, v_3+v_4\rangle$ ?

Or is one of them a subspace of the other?

Answer 1

Hint: Both are equal to $\langle v_1,v_2,v_3,v_4\rangle$.
In general, with a subspace $U$, for $\langle a_1,a_2,\dots\rangle\subseteq U$ it's enough to show that each $a_i\in U$, and equality of sets $U=V$ is implied by $U\subseteq V$ and $V\subseteq U$.