would proof by contradiction be the way to go for this problem? so i assume dim(V ∩ W) ≤ dim(v) + dim(W) - n and i also have dim(V) + dim(W) ≥ n. i can't seem to manipulate the inequalities to find a contradiction. can anyone give me a hint or two?
Given that V and W are subspaces of R^n such that dim(V) + dim(W) ≥ n, show that dim(V ∩ W) ≥ dim(v) + dim(W) - n.
so i am given dim(V) + dim(W) ≥ n and since the biggest V ∩ W can get is when V = W = R^n, then dim(V ∩ W) ≤ n. with those 2 inequalities i deduce that dim(V ∩ W) ≤ dim(V) + dim(W). then dim(V ∩ W) - n ≤ dim(V) + dim(W) - n. that's the last step i am sure about. after that step i tried a variety of manipulations to the inequalities but i was unable to show that dim(V ∩ W) ≥ dim(v) + dim(W) - n.
how do i continue on and prove this statement?