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

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

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

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.