The question is.......
Let u and v be any vectors in R^n. Prove that the spans of {u,v} and {u+v, u-v} are equal.
I'm thinking if they u and v were inverses of each other then this would work, but i'm not sure.
I'm not sure how to go about this problem. TIA
We need to show that the sets,
S={au+bv|a,b in R and u,v in R^n}
T={a(u+v)+b(u-v)|a,b in R}
Are equal.
Equivalently, S is subset of T and T is subset of S.
In set T if a=1/2 and b=1/2 then, u is in S
In set T if a=1/2 and b=-1/2 then v is in S.
Thus, their linear combination is in T.
Thus, T contains S.
In set S if a=1 and b=1 then u+v is in T.
In set S if a=1 and b=-1 then u-v is in T.
Thus, their linear combination is in T.
Thus, S contains T.
Thus, T=S