As a linear algebraist, the following interpretation is the one that helped me the most: the set of (non-necessarily pure) quantum states is in a one-to-one correspondence with all trace-1 symmetric positive-semidefinite matrices; pure states are the ones that correspond to rank-1 matrices $ | \mid \! u \rangle \langle u |$, \! \mid$, while non=pure non-pure states correspond to their (suitably scaled) linear convex combinationsof them.
So instead of thinking about vectors and their "formal linear convex combinations", you just have to think about symmetric rank-1 matrices and their linear convex combinations in the usual sense. This interpretation will also come in useful later on in your study of QM.
[EDIT: fixed two mistakes pointed out in the comments]