T:M(2x3) ---> M(2x2)

T$\displaystyle \left(\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{2 1}&a_{22}&a_{23}\end{array}\right)$ = $\displaystyle \left(\begin{array}{cc}2a_{11}-a_{12}&a_{13}+2a_{12}\\0&0\end{array}\right)$

Can anyone find the null space of this transformation, and explain why nullity(T) = 4?

R(T) = M(2x2), correct? So then shouldn't rank(T) be 4?

I'm just confused on all of these! Looking for anyone to shed some light on all of this for me.