I am studying and the textbook says to do this, though I am not sure where to start or how to prove a basis using orthogonality.
Part a) of the question asks to find the basis for $RowA$, $NulA$, $ColA$, and $NulA^T$. The matrix given is $A$.
$A$ = \begin{bmatrix} -1 & 2 & 4 & 9 & -11\\ 1 & -2 & 2 & 3 & -1\\ 3 & -6 & 7 & 11 & -5 \end{bmatrix} I've found a basis for $ColA$ to be {[-1,1,3],[4,2,7]}, a basis for $RowA (ColA^T)$ to be {[-1, 2, 4, 9, -11],[1, -2, 2, 3, -1]}, a basis for $NulA$ to be {[2,1,0,0,0],[1,0,-2,1,0],[-3,0,2,1]}, and a basis for $NulA^T$ to be {[-1/6, -19/6, 1]}.
The next part of the question, part b), asks to check your answers using orthogonality conditions, but it is unclear as to what these conditions are. I am hoping for some guidance.