Suppose {u,v} are linearly independent, and that {u+v,u-v} are not,

Then there exist a, b in R, at least one of which !=0, such that:

a(u+v) + b(u-v)=0,

so:

(a+b)u + (a-b)v=0

but now at least one of (a+b) and (a-b) !=0, so we conclude that

u and v are linearly dependent - a contradiction.

So we conclude that if {u.v} are linearly independent then so are {u+v,u-v}.

The second half of the proof proceeds like the above but starting from

the assumption that {u+v,u-v} are linearly independent and that [u,v}

are not.

RonL