Finding the intersection of subspaces, and addition of subspaces

[minitejpar]

Heres the question:
Let {u,v,w} be a linearly independent set of vectors of R^4. Let E = span{u,2v} and F=span{w,v}. Find EnF and E + F.

i really have no idea other than i guess if 1/2u=w and v=v, then the EnF can be defined by that, but I'm not sure if that is right! :(

 

[lanedance]

if u & v are linearly independent, what does span{u,v} represent?

now does span{u,2v} contain v?

 

[minitejpar]

span{u,v} represents all the combinations of u and v. (at least that's what i understand).

and yes, span(u,2v) does contain v, and so can i multiply 2v by 1/2? and then have span{u,v} instead of span{u,2v}?

if this is the case, then how do i find the intersect if F=span{w,v} seeing as w is not u..?

 

[vela]

span{u,v} represents all the combinations of u and v. (at least that's what i understand).

Can you be a bit more specific as to what you mean by "combination"?

and yes, span(u,2v) does contain v, and so can i multiply 2v by 1/2? and then have span{u,v} instead of span{u,2v}?

if this is the case, then how do i find the intersect if F=span{w,v} seeing as w is not u..?

When a vector x is in E ∩ F, that means x∈E and x∈F. So you're looking for all vectors that can be expressed as a combination of u and 2v (so that it's in E) and a combination of w and v (so that it's in F). So again, it depends on what exactly you mean by "combination."

 

[lanedance]

span{u,v} represents all the combinations of u and v. (at least that's what i understand).

span{u,v} = the set containing all linear combinations of u & v, so any vector w with:
w = a.u + b.v
for any real scalars a,b

think of u & v as vectors, what does span{u,v} represent geometrically?

note that
0 = 0.u + 0.v is in span{u,v}

and yes, span(u,2v) does contain v, and so can i multiply 2v by 1/2? and then have span{u,v} instead of span{u,2v}?

so then can you show

span{u,v} = span{u,2v}

if this is the case, then how do i find the intersect if F=span{w,v} seeing as w is not u..?

 

[Outlined]

Your very very first step should be E = span{u, 2v} = span{u, v}. This makes life easier.

 

[Mamooie312]

E + F = span{u,2v) + span{w,v} = au + 2bv + cw + dv = span(u,v,w) I think. Not comepletely sure.