Special solutions to Ax = 0 I solved most of it, just not sure about one point.
The problem statement, all given variables and data
Suppose A is the matrix shown below:
$$
        \begin{pmatrix}
        0 & 1 & 2 & 2 \\
        0 & 3 & 8 & 7 \\
        0 & 0 & 4 & 2 \\
        \end{pmatrix}
$$
Find all special solutions to Ax = 0. 
Attempt at a solution
So after some elimination, I acquired the matrix below.
$$
        \begin{pmatrix}
        0 & 1 & 0 & 1 \\
        0 & 0 & 1 & -1/2 \\
        0 & 0 & 0 & 0 \\
        \end{pmatrix}
$$
According to what I read online, there must be special solutions as many as the number of free variables. You go through the free variables one by one, making one of them equal to 1 and the rest equal to 0. So my first free solution is below:
Let $x_4 = 1$. $\Rightarrow x_1 = ?$ (there is no $x_1$), $x_2 = -1$, $x_3 = -\frac{1}{2}$ and $x_4 = 1$
Let $x_1 = 1$ (but there is no $x_1$) $\Rightarrow x_1 = ?$, $x_2 = 0$, $x_3 = 0$, $x_4 = 0$
I am not entirely sure if I concluded the answer correctly. I would appreciate if someone could wrap it up. If I don't have a variable in my system at all, like $x_1$ here, then what exactly do I put in its place? How do I represent it?
 A: Even though one may say $x_1$ is not contributing anything but this very fact makes it a free variable. So in your case both $x_1$ and $x_4$ are free variables. The remaining are pivot variables. So what you are doing is correct. You can have $x_1=1$ and $x_4=0$ for the second special solution.
Note:
Another way to think about it is that $Ax$ is also the linear combination of columns of $A$ with coefficients from the vector $x$. So in effect you have a situation like this
$$
x_1
\begin{bmatrix}
0\\0\\0
\end{bmatrix}
+
x_2
\begin{bmatrix}
1\\3\\0
\end{bmatrix}
+
x_3
\begin{bmatrix}
2\\8\\4
\end{bmatrix}
+
x_4
\begin{bmatrix}
2\\7\\2
\end{bmatrix}
=
\begin{bmatrix}
0\\0\\0
\end{bmatrix}
$$
Now by setting $x_1=1$ and the rest of them as $0$, you get a solution to this homogeneous system.
A: I found a good treatment of determining the null space of a matrix A (containing all vectros x with A*x=0) in J. L. Goldberg, Matrix Theory with Applications, pp.123-128, McGraw-Hill, 1991. See also http://en.wikipedia.org/wiki/Kernel_(linear_algebra)
