Matthew S. answered • 03/12/20

PhD in Mathematics with extensive experience teaching Linear Algebra

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)