vector spaces - Span and Linear Independence of two sets

Let $V$ be a vector space over a field $F$, let $v_1, v_2, v_3 ∈ V$, and let
$u_1 = v_1, u_2 = 2v_1 + v_2$, and $u_3 = 3v_1 + 2v_2 + v_3$,
which are all elements of $V$.
Prove the following:
• If $\{v_1, v_2, v_3\}$ is linearly independent, then $\{u_1, u_2, u_3\}$ is linearly independent.
• If $\{v_1, v_2, v_3\}$ spans $V$, then $\{u_1, u_2, u_3\}$ spans $V$

Note: I figured out the first part by plugging in the values for $\{u_1, u_2, u_3\}$ in a linear combination and equating it to zero, proving both the sets are linearly independent. Not sure how to prove that these vectors span the vector space $V$.