I need to prove that these statements are true or false.
1. It is possible to find a pair of two-dimensional subspaces S and T of R³ such that S ∩ T = {0}
2. If S and T are subspaces of a vector space V, then S U T is a subspace of V.
3. If S and T are subspace of a vector space V, then S ∩ T is a subspace of V.
I understand that the two closure identities need to be met in order to be a subspace. However the intersection and union properties are throwing me off. Can anyone help me with this? Thank you.
No. Remember the dimension of over is .
This means that you cannot have a set with more than elements which are linearly independent.
Let and .2. If S and T are subspaces of a vector space V, then S U T is a subspace of V.
Do the following steps.
1)Show are sub-spaces.
2)Show is not closed, i.e. there are so that
3)Conclude that is not a sub-space.
That makes sense now, thanks!
s = (1,0,1) and u = (1,1,0) then sUv = (2,2,1) which does not belong to either s or u. Does that work?Let and .
Do the following steps.
1)Show are sub-spaces.
2)Show is not closed, i.e. there are so that
3)Conclude that is not a sub-space.