linear dependence/ independence?

[Nupur]

[Amit Goyal]

Suppose u, v, w are three vectors in 3-dimensional space, and are linearly dependent, then the following is true:
a(u) + b(v) + c(w) = 0 for some scalars a, b, c (not all zero) -- (1)

We are asked to check for the (in)dependence of vectors u + v, v + w, u + w. Clearly, it follows from (1) that
(a + b - c)/2 (u + v) + (b + c - a)/2 (v + w) + (a + c - b)/2 (u + w) = 0

We will now show that if at least one of a, b and c is non-zero, then at least one of (a + b - c)/2, (b + c - a)/2, (a + c - b)/2 is non-zero.
Proof: We will prove its contra-positive. Suppose (a + b - c)/2, (b + c - a)/2, (a + c - b)/2 are all zero, then
a + b = c
b + c = a
a + c = b
implying a = b = c = 0.

Thus (1) implies (a + b - c)/2 (u + v) + (b + c - a)/2 (v + w) + (a + c - b)/2 (u + w) = 0
for scalars (a + b - c)/2, (b + c - a)/2, (a + c - b)/2 where not all are zero.
Therefore, u + v, v + w, u + w are linearly dependent (option (c)).

[Ann Joseph]

There are many situation, when we might want to test if the set of vectors are linearly dependent or not. You give a wonderful explanation in your proof.
Thank you.
Tips for writing college essays