How to tell if a matrix of a certain rank contains a certain kernel and image? I'm struggling with the following question
True or False: There exists a 3x4 matrix A of rank 2 such that ker(A) contains the vector v = $$
    \begin{bmatrix}
    1\\
    1\\
    1\\
    1\\
    \end{bmatrix}
$$
and image(A) contains the vector w = $$
    \begin{bmatrix}
    1\\
    1\\
    1\\
    \end{bmatrix}
$$
I'm not sure how to go about this without coming up with matrices that would possibly fit these conditions. How else can you go about this question?

How about $\begin{pmatrix}1&-1&0&0\\1&-1&0&0\\1&0&-1&0\end{pmatrix}$?
Not sure about how else to go about it, than providing an example. 
