Proof of linear independence

**problem:**

(a)

Prove that the four vectors

$\displaystyle x=(1,0,0),\;y=(0,1,0),\;z=(0,0,1)\;u=(1,1,1) $ in $\displaystyle \mathbb{C}^3$ form a linearly dependent set, but any three of them are linearly independent.

(b)

$\displaystyle x(t)=1,\;y(t)=t,\;z(t)=t^2,\;u(t)=1+t+t^2 $, prove that $\displaystyle x,y,z,u$ are linearly dependent, but any three of them are linearly independent.

**attempt:**

(a) To prove that the four vectors are linearly dependent, I guess I could set up four linear equations and solve them.

But I'm thinking it should be enough to show that:

$\displaystyle x+y+z-u=0$.

The thing I do not understand is the "any three of them are linearly independent" part.

How do I go about proving that without checking all combinations?

(b) $\displaystyle x+y+z-u=0$. Again, I don't know how to prove the "any three of them" part without checking all combinations.

Thanks!