Let x be a generic vector (a,b,c,d). So, the product A*x is[tex]\left(\begin{array}{cccc}0&0&0&0\\2&2&-2&-2\end{array}\right)\left(\begin{array}{c}a\\b\\c\\d\end{array}\right) = \left(\begin{array}{c}0\\2a+2b-2c-2d\end{array}\right)[/tex]
We want this vector to be zero. The first entry is already zero, so we have to impose
[tex]2a+2b-2c-2d = 0 \iff a+b-c-d=0[/tex]
This is one equation in four variables, and thus we can fix three values and derive the fourth in terms of the previous three. In particular, if we fix a, b, c and derive d, we have
[tex]a+b-c-d=0 \iff d = a+b-c[/tex]
So, a generic solution to the system looks like
[tex]\left(\begin{array}{c}a\\b\\c\\a+b-c\end{array}\right) = a\cdot \left(\begin{array}{c}1\\0\\0\\1\end{array}\right) + b\cdot \left(\begin{array}{c}0\\1\\0\\1\end{array}\right) + c \cdot \left(\begin{array}{c}0\\0\\1\\-1\end{array}\right)[/tex]
So, the three spanning vectors are (1, 0, 0, 1), (0, 1, 0, 1) and (0, 0, 1, -1).
