I need to know if my work is right.
T$\begin{bmatrix}a11 & a12 & a13\\a21 & a22 & a23\\a31 & a32 & a33\end{bmatrix}$=$\begin{bmatrix}a11+a12+a13 & a21+a22+a23\\ a31+a32+a33 & 0\end{bmatrix}$
Question: Find bases for the nullity and rank
Attempt:
Basis for N(T)={(-1,0,1), (-1,1,0), (-1,0,1),(-1,1,0), (-1,0,1), (-1,1,0) } Dim(N(T))=6
To find the rank, I used the dimension theorem. Dim V=nullity T + Rank T.
Dim V=9 and Nullity T= 6 so 9-6= Rank T = 3
Is the basis for the range just the standard basis for M2x2?