Determine if the following set of vectors is linearly independent:

$$\left[\begin{array}{r}2\\2\\0\end{array}\right],\left[\begin{array}{r}1\\-1\\1\end{array}\right],\left[\begin{array}{r}4\\2\\-2\end{array}\right]$$

I've done the following system of equations, and I think I did it right... It's been such a long time since I did this sort of thing...

Assume the following: \begin{equation*} a\left[\begin{array}{r}2\\2\\0\end{array}\right]+b\left[\begin{array}{r}1\\-1\\1\end{array}\right]+c\left[\begin{array}{r}4\\2\\-2\end{array}\right]=\left[\begin{array}{r}0\\0\\0\end{array}\right] \end{equation*} Determine if $a=b=c=0$: \begin{align} 2a+b+4c&=0&&(1)\\ 2a-b+2c&=0&&(2)\\ b-2c&=0&&(3) \end{align} Subtract $(2)$ from $(1)$: \begin{align} b+c&=0&&(4)\\ b-2c&=0&&(5) \end{align} Substitute $(5)$ into $(4)$, we get $c=0$.

So now what do I do with this fact? I'm tempted to say that only $c=0$, and $a$ and $b$ can be something else, but I don't trust that my intuition is right.