I'm trying to wrap my head around the meaning of different dimensions in lin alg. Consider the matrix:
$\left[ \begin{array}{ccc} 1 & -1 \\ 2 & -2\\ 3 & -3 \end{array} \right] $
I've found that $C(A) = $ span of $\left[ \begin{array}{ccc} 1 \\ 2\\ 3 \end{array} \right] $ and $N(A) = $ span of $\left[ \begin{array}{ccc} 1 \\ -1 \end{array} \right] $
and similarly I can find the $C(A^T)$ and $N(A^T)$. However, what if I'm asked for the $R(A^T)$ and $R(A)^\perp$ i.e. the orthogonal complement of the row space of A.
I believe the column space of $A$ is the row space of $A^T$, so that's fine. The row space of $A$ is the column space of $A^T$, but to find its orthogonal complement, do I find the null space of the column space of $A$ i.e. the null space of $A$?
My question boils down to:
Is $N(A)$ = orthogonal complement of the row space of $A$?
How do they all tie together? I know the number of vectors given by the dimension's theorem, i.e. $r$, $n-r$, but I'm not sure how they fit in intuitively.
Thanks in advance.