A fact about linearly independent triples

Let F be a field and let V be an F-vector space. Suppose T = \{x,y,z\} \subseteq V is linearly independent. Show that S = \{x+y,x-z,y-z\} is also linearly independent.


Suppose a(x+y) + b(x-z) + c(y-z) = 0; then (a+b)x + (a+c)y - (b+c)z = 0. Since T is linearly independent, a+b = a+c = b+c = 0. In particular, b = c and b = -c; thus b = c = 0, and so a = 0 as well. Thus S is linearly independent.

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