I'm trying to give a combinatorial proof of the q-Chu-Vandermonde identity: $\binom{m + n}{k}_{q} =\sum_{j} \binom{m}{k - j}_{q} \binom{n}{j}_{q} q^{j(m-k+j)}.$
I understand that the LHS counts the number of dimension $k$ subspaces of a vector space of dimension $m + n$ of a Finite Field with $q$ elements. I see how the traditional committee forming argument for the Chu-Vandermonde identity could be applied since the RHS is the sum over all possible combinations of dimension $j$ subspaces of a vector space of size $n$ and dimension $k-j$ subspaces of a vector space of size $m$, but I am not sure how to make this proof more rigorous specifically regarding the algebra aspect.