Find the matrix of a linear transformation given kernel

Find the matrix associated to a linear function:

$ F: \Bbb R^3 \to \Bbb R^4 $

$ \ker F=\left\{(x,y,z) \in \Bbb R^3 \mid x - 2y + z = 0\right\} $

and

$F((0,1,1))=(-1,-3,0,1)$

It's a multiple choice exercise, the possible answers are:

$ a = \begin{pmatrix} 1 & -2 & 1 \\ 2 & -4 & 2 \\ 0 & 0 & 0 \\ 3 & -6 & 3 \end{pmatrix}$

$ b = \begin{pmatrix} 1 & -2 & 1 \\ 2 & -4 & 2 \\ 0 & 0 & 0 \\ -3 & 6 & -3 \end{pmatrix}$

$ c = \begin{pmatrix} 1 & -2 & 1 \\ 3 & -6 & 3 \\ 0 & 0 & 0 \\ -1 & 2 & -1 \end{pmatrix}$

$ d = \begin{pmatrix} 2 & -2 & 1 \\ 4 & -6 & 3 \\ 0 & 0 & 0 \\ 2 & 2 & -1 \end{pmatrix}$

I don't know ho to proceed, can someone help me?