**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:

a**v + **b**w** + c**u** = 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:

a**v + **b**w** + c**u** = 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.