Linear Algebra: intersection of subspaces

 P: 4 1. The problem statement, all variables and given/known data I'm working on a problem that involves looking at the dimension of the intersection of two subspaces of a vector space. 2. Relevant equations $M \subset V$ $N \subset V$ dim(M $\cap$ N) $[\vec{v}]_{B_M}$ is the coordinate representation of a vector v with respect to the basis for M 3. The attempt at a solution I reformulated M $\cap$ N in a bunch of different ways that would be too long to copy down here, but I finally came to this (which may or may not be useful to me in my larger problem but I'm wondering if it is valid itself): $\vec{v}$ is itself, so it must have the same dimension in both M and N, and since the bases are ordered, for each $\vec{b}_{Mi}$ in $B_M$ for which the corresponding scalar is not zero in the linear combination of elements of $B_M$ equal to $\vec{v}$, and each $\vec{b}_{Nj}$ in $B_N$ for which the corresponding scalar is not zero in the linear combination of elements of $B_N$ equal to $\vec{v}$, if i=j then $\vec{b}_{Mi}$ and $\vec{b}_{Nj}$ are dependent and $[\vec{v}]_{B_M}$ has zeros in the same places as $[\vec{v}]_{B_N}$ but there is a major problem here with the fact that we may have dimM ≠ dimN