Thread: Dimension problem

Let V be a vector space and U and W subspaces of V.

Show that:
is equivalent to: U≠W

I think one direction of the equivalence is easy to prove:
If U = W then U+W=2U=U≠V.
I hope this is right. I don't know how to prove the other direction though.

Any help would be greatly appreciated.

Originally Posted by durrrrrrrr

I think one direction of the equivalence is easy to prove:
If U = W then U+W=2U=U≠V I hope this is right.

Right, but don't write . Simply

I don't know how to prove the other direction though.

Suppose then,

or

Could you continue?.

Regards.

Fernando Revilla

Originally Posted by durrrrrrrr

Let V be a vector space and U and W subspaces of V.

Show that:
is equivalent to: U≠W

I think one direction of the equivalence is easy to prove:
If U = W then U+W=2U=U≠V.
I hope this is right. I don't know how to prove the other direction though.

Any help would be greatly appreciated.

There is another way which gives us more information:

Spoiler:

Note that from a common theorem we have that

Note though that since we must have that and so by we know that . But, since we know that . Thus, putting these two together gives

Thus, equals either or . And so plugging this into gives

But note that and (the first is clear, and the second follows since and a subspace which has the same dimension as the ambient space must be the full space). Thus, this may be rewritten as

Thus, if then is an -dimensional subspace of and thus equal to and if then is an -dimensional subspace and not equal to . The conclusion follows.