If the set {v, w, x} is linearly independent, so is {v+w, v+x, w+x}

Question

Let v, w, x ∈ Rⁿ be vectors.

Prove that if the set of vectors {v, w, x} is linearly independent, then so is the set {v + w, v + x, w + x}.

Answer 1

Let a, b and c be scalars such that

a(v + w) + b(v + x) + c(w + x) = 0

We want to show that a, b and c are all 0.

Expanding the left hand side gives you

(a+b)v + (a+c)w + (b+c)x = 0

a+b = 0, a+c = 0 and b+c = 0 because v, w and x are linearly independent

Subtracting the second equation from the first: 0 = (a+b) - (a+c) = b-c. Add this new equation to the third equation: 2b = 0, so b = 0. Then a and c are also zero (from first and third equations)