Okay, we have that $\{|a_i\rangle\}_{i=1}^n$ is a set of vectors spanning a vector space $V$. Also, $T\in L(V,W)$ is surjective, where $L(V,W)$ is the set of linear transformations (functionals) from $V$ to $W$.
I need to show that, given the above, $\{T$ $|a_i\rangle\}_{i=1}^n$ spans $W$.
And I don't know where to start.
HINT: Every $w\in W$ is the image under $T$ of some $v\in V$, and $v=\sum_{k=1}^nc_k|a_k\rangle$ for some constants $c_k$, $k=1,\dots,n$, so $w=Tv=T\sum_{k=1}^nc_k|a_k\rangle=\;?$