Q:  Find two vectors v and w such that the three vectors u = (1,-1,-1), v and w are linearly independent independent.

A:  By definitions, vectors v, w, and u are linearly independent if:

av + bw + cu = 0 → a = b = c = 0

There are so many vectors to pick for v and w (infinite choices).  You just need to pick two that will work and then prove that it works!

So, I claim that v = (1, 0, 0) and w = (0, 0, 1) will give us three linearly independent vectors.

Here comes my proof:

av + bw + cu = 0

a(1, 0, 0)+ b(0, 0, 1) + c(1, -1, -1) = 0

So, we know that

(1)  1a + 0b + 1c = 0

(2)  0a + 0b – 1c = 0

(3)  0a + 1b – 1c = 0

Now simplify:

(1)  a + c = 0

(2)  -c = 0

(3)  b – c = 0

From (2) we have that c = 0

So in (3) b – c = 0 → b – 0 = 0 → b = 0

And (1) a + c = 0 → a + 0 = 0 → a = o

Therefore, a = b = c = 0.  Vectors v, w and u are linearly independent.