You might guess, by analogy with the formula for the number of elements in the union of three subsets of a finite set, that if U1,U2,U3 subspaces of a finite-dimensional vector space, then
dim(U1+U2+U3)=−+dimU1+dimU2+dimU3dim(U1∩U2)−dim(U1∩U3)−dim(U2∩U3)dim(U1∩U2+∩U3)
Prove this or give a counterexample.
We know subspace addition is associative (see 1c/17) so we can prove this by applying the two subspace case repeatadly
dim((U1+U2)+U3)=−dim(U1+U2)+dimU3dim((U1+U2)∩U3)
Since (U1+U2)∩U3=(U1∩U3)+(U2∩U3) we can write
dim((U1+U2)∩U3)=dim((U1∩U3)+(U2∩U3))=dimU1∩U3+dimU2∩U3−dimU1∩U2∩U3
In the last line I used
(U1∩U3)∩(U2∩U3))=U1∩U2∩U3
Anyway, combine this with the previous result to get
dim(U1+U2+U3)=−+dimU1+dimU2+dimU3dim(U1∩U2)−dim(U1∩U3)−dim(U2∩U3)dim(U1∩U2+∩U3)
As desired