- #1
Dansuer
- 81
- 1
Homework Statement
[itex]V[/itex] is a vector space with dimension n, [itex]U[/itex] and [itex]W[/itex] are two subspaces with dimension k and l.
prove that if k+l > n then [itex]U \cap W[/itex] has dimension > 0
Homework Equations
Grassmann's formula
[itex]dim(U+W) = dim(U) + dim(W) - dim(U \cap W)[/itex]
The Attempt at a Solution
Suppose k+l >n.
Suppose that [itex]dim(U \cap W) \leq 0[/itex]
since the dimension can't be negative [itex]dim(U \cap W) = 0[/itex]
then Grassman formula reduces to
[itex]dim(U+W) = dim(U) + dim(W)[/itex]
[itex]dim(U+W) = k +l > n[/itex]
this is a contraddiction because [itex]U+W[/itex] has dimension grater than the dimension of his enclosing space.
is this a valid proof?
Last edited: