Determining the nullspace and range of a linear transformation

Question

T is the linear transformation from M2x3 to M2x2:

T$\begin{bmatrix}a & b & c \\ d & e & f \end{bmatrix}$=$\begin{bmatrix}a+b & b+c\\ d+e & c+f\end{bmatrix}$

Question : Assuming T is a linear transformation, find bases for N(T) and R(T)

Attempt :

I tried to solve using a 4x6 matrix and I got a=-f, b=f, c=-f , d=-e. I set d=0 and f as a free variable (1) and get a=c=-1, b=f=1, , d=e=0 . Then I set f=0 and e as the free variable (!) and I got a=b=c=f=0, d=-1, e=1. After this Im lost. Dont know what to do next.

Answer 1

Note that

$$[T]=\begin{bmatrix}1&1&0&0&0&0\\0&1&1&0&0&0\\0&0&0&1&1&0\\0&0&1&0&0&1 \end{bmatrix}$$

which has $rank=4$ then refer to Rank–nullity theorem.