Dimension of intersection of subspaces proof

[Dansuer]

Homework Statement

V is a vector space with dimension n, U and W are two subspaces with dimension k and l.
prove that if k+l > n then U \cap W has dimension > 0

Homework Equations

Grassmann's formula

dim(U+W) = dim(U) + dim(W) - dim(U \cap W)

The Attempt at a Solution

Suppose k+l >n.
Suppose that dim(U \cap W) \leq 0

since the dimension can't be negative dim(U \cap W) = 0

then Grassman formula reduces to

dim(U+W) = dim(U) + dim(W)
dim(U+W) = k +l > n

this is a contraddiction because U+W has dimension grater than the dimension of his enclosing space.

is this a valid proof?

 

[HallsofIvy]

What, exactly, are you trying to prove?

 

[Dansuer]

that dim(U \cap W) > 0
since you are asking me that question do i have to assume that my proof is wrong?

EDIT: ops

I have to prove that if k+l > n then dim(U \cap W) > 0