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.