Question

let v1=(1,0,0) and v2=(0,1,0) and h=(s,s,0): s in R. Then every vector in H is a linear combination of v1 and v2 because (s,s,0)= sv1+sv2.
Is {v1,v2} a basis for H? let v1=(1,0,0) and v2=(0,1,0) and h=(s,s,0): s in R. Then every vector in H is a linear combination of v1 and v2 because (s,s,0)= sv1+sv2.
Is {v1,v2} a basis for H? @Mathematics

Answer 1

Yes, since any vector in H can be made from v1 and v2, and v1 and v2 are linearly independent.

Answer 2

according to the book it says neither v1 nor v2 is in H so {v1 v2} cannot be a basis for H. In fact {v1 v2} is a basis for the plane of all vectors of the form (c1, c2, 0) but H is only a line.

I have no idea what that means though

Answer 3

I see now. The book is correct. Since a basis must be a subset of H, but v1, and v2 can't be in H, v1 and v2 can't form a subset of H, and thus can't be a basis.