Proof of the spans of u and v vectors in R^n I am having trouble putting togethe

Question

Proof of the spans of u and v vectors in R^n I am having trouble putting together a proof for the following question.
"Let u & v be any vectors in Rn. Prove that the spans of {u,v} and {u+v, u-v} are equal."

Any help would be greatly appreciated.

Thanks Proof of the spans of u and v vectors in R^n I am having trouble putting together a proof for the following question.

Explanation / Answer

suppse w is an arbitrary vector in the span of { u,v} that means w = au + bv for some scalars where atleast one of a or b is non zero. the same w can be written as the linear combination of u+v and u-v such that the atleast one of the scalars is not zero. suppose w = c(u+v) + d(u-v) ==> w = (c+d) u + ( c-d)v comparing this linear combination with the above, we have c+d = a and c - d = b so, c = (a+b)/2 and d = (a-b)/2 while atleast one of a and b is non zero, we can confirm that at least one of a+b/2 or a-b/2 is non zero. that means at least one of c or d is non zero such that w = c(u+v)+d(u-v) so, w is in the linear span of u+v and u-v therefore, L{u,v} is a subset of L{u+v,u-v} reversing the argument above, we can show L{u+v,u-v} is a subset of L{u,v} thus L{u+v,u-v} = L{u,v} Hence the theorem.

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