1. ## Linear dependence/independence

If it's in $r^3$, can you still do it if there is 4 vectors but each have x y and z?

If it's in $r^3$, can you still do it if there is 4 vectors but each have x y and z?
Do you want to ask whether more than n vectors from $\scriptstyle\mathbb{R}^n$ (here n=3) can be linearly independent? - If so, the answer is no, because a system of n (here 3) homogeneous linear equations in more than n variables (here 4) always has non-trivial solutions.
Or, in fewer words: 4 vectors from $\scriptstyle\mathbb{R}^3$ cannot be linearly independent (they are necessarily linearly dependent).

3. because a system of n (here 3) homogeneous linear equations in more than n variables (here 4) always has non-trivial solutions.
Unless the system is inconsistent.

4. Originally Posted by Ackbeet
Unless the system is inconsistent.
Well, no: a homogeneous linear system cannot be inconsistent, since it always has the trivial solution.

5. True. You did say homogeneous, and I was thinking non-homogeneous.