Is W1\cap W2 a Vector Space if dim(W1)=1 and dim(W2)=2?

[transgalactic]

there are two W1 and W2 of F^3 space
dim(W1)=1
dim(W2)=2

prove or desprove that:

W1\cap W2={0} is the vector space
??

there could be a case where W2 includes W1 then there intersection is not the 0 space
correct??

 

[Quantumpencil]

If there are no other constraints on the problem then yeah.

Take a plane going through the origin and a line contained on that plane in R^3.

That seems like a silly problem though. Do you know anything else like

W1+W2 = W3 (direct sum?)

 

[transgalactic]

can you give an actual example
??

 

[Quantumpencil]

I am just saying your counterexample (or one like it) works as long as there are no other restrictions on the problem.

If we require that W1+W2 = F^3, then it is true, because one space cannot contain the other.

 

[transgalactic]

if W2 include W1
and we have another subspace W3
which
dim W3=1
then dimW2+dimW3=3

what is the problem in that??

 

[Quantumpencil]

Nothing, that's tenable.