<s>[INST] For c = 3 find the complete solution $x$ to $Ax = \begin{bmatrix}1\\c\\0\end{bmatrix} $ the matrix is: $$\color{green}{ \begin{bmatrix}1&1&2&4\\3&c&2&8\\0&0&2&2\end{bmatrix} }\cdot$$
reducing the matrix gives me:
$$\color{green}{ \begin{bmatrix}1&1&2&4\\0&c-3&-4&-4\\0&0&2&2\end{bmatrix} }\cdot$$
I already calculated the solution for the nullspace of A
$$\color{green}{ \begin{bmatrix}-2\\0\\-1\\1\end{bmatrix} }\cdot$$
the solution is one vector + the solution for the nullspace help me find that first vector by inspection
 [/INST] $\begin {pmatrix}0\\1\\0\\0\end {pmatrix} $ works nicely as a particular solution, and the solution set is thus $$\{\begin {pmatrix}0\\1\\0\\0\end {pmatrix}+s\cdot \begin {pmatrix}-2\\0\\-1\\1\end {pmatrix}+t\cdot \begin {pmatrix}1\\-1\\0\\0\end {pmatrix}\mid s,t\in\Bbb F\}$$.
That's the null space is actually $\color{red}2$-dimensional.
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