I understand that with most vectors to express them as a linear combination you can use the sort of method shown Here but with the zero vector surely everything in the $ w$ column is a $0$ so you can't get the $x$,$ y$ and $z$ values, only trivial combinations.
Any guidance on how to do this would be greatly appreciated. I just want to know how to do the first step really not the whole thing. Thanks in advance.
Edit: Trying to use Gauss jordan algorithm for it. Edit: I have been given the three vectors to use but I don't want the full answer just how to do the next step so I can do it myself
You keep saying that that zero vector causes problems, so you can't get anywhere. In fact those zeroes are no problem at all.
A similar problem (which is really more than I should provide here, since there are similar problems worked out earlier in the book): Express the zero vector as a non-trivial linear combination of $(1,2)$ and $(2,4)$:
We want a non-trivial solution to $x(1,2)+y(2,4)=(0,0)$. That says $$\begin{align}x+2y&=0 \\2x+4y&=0.\end{align}$$
You are supposed to know a procedure for finding the general solution to any system of linear equations. If you simply apply that algorithm you find the general solution is $$x=-2t, y=t,$$where $t$ is a parameter. (If you don't know how I got that solution you're in big trouble in your linear algebra course! You need need need to seriously review a few earlier sections on solving linear equations!) If you set $t=0$ you get $x=y=0$, not the solution you want. But if you set $t=1$ you get $x=-2, y=1$, a non-trivial solution.