How to solve this to find the Null Space

Write the system

$$\left\{\begin{array}{rcl} x-2y+2z+4t& =& 0 \\ z+t & = & 0\end{array}\right.$$

as

$$\left\{\begin{array}{rcl} x+2z& =& 2y-4t \\ z & = & -t\end{array}\right.$$ and solve it. You get,

$$\left\{\begin{array}{rcl} x& =& 2y-2t \\ z & = & -t\end{array}\right.$$ That is, $$(2y-2t,y,-t,t)$$ is an element of the null space for any $y,t.$ Now, look for two linearly independent vectors.

(Note that the kernel has dimension $2.$ So the system has infinitely many solutions that have to depend on two parameters.)

Do also backwards elimination: $$ \begin{bmatrix} 1 & -2 & 2 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \to \begin{bmatrix} 1 & -2 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$ Now your equations read $$ \begin{cases} x_1=2x_2-2x_4\\ x_3=-x_4 \end{cases} $$ You get two linearly independent vectors in the null space by setting $x_2=1, x_4=0$ and $x_2=0,x_4=1$, so the basis is given by the two linearly independent vectors $$ \begin{bmatrix} 2\\ 1\\ 0\\ 0\\ \end{bmatrix} \qquad \begin{bmatrix} -2\\ 0\\ -1\\ 1 \end{bmatrix} $$ The first corresponds to $x_2=1$ and $x_4=0$, the second to $x_2=0$ and $x_4=1$.

Just to make the answer a bit more algorithmic: a "pivot entry" is an entry which is the first non-zero entry in its row. A "pivot column" is a column containing a pivot entry. In your matrix, columns 1 and 3 are pivot columns.

Name the variables after the columns as you did (so, $x, y, z, w$). Then the ``free variables'' are the ones that don't occur in pivot columns. In your case, these are $y$ and $w$. The remaining rows of the matrix express the bound variables in terms of the free variables.