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iloveannaw
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Homework Statement
Given is a vector space (V,+,k) over kn with n > 1. Show that with
[tex]W \subseteq V[/tex], [tex]U \subset V [/tex]and [tex]dim(U) = n - 1[/tex]
[tex]dim(W \cap U) \geq dim(W) - 1[/tex]
Homework Equations
[tex]dim(W+U) +dim(W \cap U) = dim(W) +dim(V)[/tex]
The Attempt at a Solution
[tex]dim(V) = n [/tex][tex]dim(W) \leq dim(V)[/tex]
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) [tex]dim(U) < dim(W) \leq dim(V)[/tex]
b) [tex]dim(U) \leq dim(W) < dim(V)[/tex]
c) [tex]dim(W) < dim(U) \leq dim(V)[/tex]
the result of a and b are the same [tex] dim(W \cap U) = dim(W) [/tex]
in all cases [tex] dim(W \cap U) \geq dim(W) - 1[/tex]
but how can you show this in a nice clean way?
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