The problem involves proving that the intersection of two subspaces, W1 and W2, in R^n is also a subspace. Participants are exploring the properties of vector spaces and the implications of closure under addition and scalar multiplication.
The discussion is ongoing, with participants exploring the implications of closure properties and seeking to express their reasoning in a formal mathematical context. Guidance has been offered on how to structure the proof, but no consensus has been reached on the final argument.
Participants express uncertainty about how to generalize their understanding beyond specific cases in R^2 and R^3, indicating a need for clarity on the definitions and properties of subspaces.
VeeEight said:Let W be the intersection of W1 and W2. Is the vector space W closed under addition and scalar multiplication?
Well, if not, you're going to have a tough time proving it.mlarson9000 said:Wouldn't it have to be?
Take a couple of arbitrary vectors u1 and u2 in W, and show that their sum is also in W. Then take an arbitrary scalar s, and show that su1 is in W. That's how you would do it it "math speak."mlarson9000 said:If W is in both W1 and W2, which are both subspaces and therefore closed under addition and scalar multiplication, wouldn't W be also?
mlarson9000 said:If so, I still don't know exactly how to say that in Math speak.
Suppose u and v are in W. Then u and v are both is W1 and, since W1 is a subspace, u+ v is in W1. Also u and v are both in W2 ...mlarson9000 said:Wouldn't it have to be? If W is in both W1 and W2, which are both subspaces and therefore closed under addition and scalar multiplication, wouldn't W be also? If so, I still don't know exactly how to say that in Math speak.