Dimension of vector space intersect with one proper subset

[iloveannaw]

Homework Statement

Given is a vector space (V,+,k) over kⁿ with n > 1. Show that with

W \subseteq V, U \subset Vand dim(U) = n - 1

dim(W \cap U) \geq dim(W) - 1

Homework Equations

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

The Attempt at a Solution

dim(V) = n
dim(W) \leq dim(V)

dim(W+U) is equal to the dimension of the 'smallest' subset (depending whether dim(W) is less than or greater than dim(U)).

the way i see it the are three distinct cases. Either
a) dim(U) < dim(W) \leq dim(V)
b) dim(U) \leq dim(W) < dim(V)
c) dim(W) < dim(U) \leq dim(V)

the result of a and b are the same dim(W \cap U) = dim(W)

in all cases dim(W \cap U) \geq dim(W) - 1

but how can you show this in a nice clean way?

 

[micromass]

You have the dimension theorem

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

Now plug in dim(U)=n-1. Now there are two cases to consider: dim(U+W)=n-1 or n.