Linear Algebra: intersection of subspaces

[TheTangent]

Homework Statement

I'm working on a problem that involves looking at the dimension of the intersection of two subspaces of a vector space.

Homework 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

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

 

[sunjin09]

first imaging a basis for the intersection {b1,b2,...,bp}, then expand it to a basis of M, {b1,b2,...,bp,m1,m2,...,mq},
also expand it to a basis for N, {b1,...,bp,n1,n2,...,nr}, dimension of the intersection is p, dim(M)=p+q, dim(N)=p+r.

 

[TheTangent]

beautiful