Q:

Why the following $dim(N(A)) + dim(C(A^T)) = n$ is true?

I was studying about row, column, null and left null spaces, and one thing I don't understand is why the dimension of the null space of a matrix $+$ the dimension of a the row space is equal to the number of columns?
In other words, why the following is true:
$$dim(N(A)) + dim(C(A^T)) = n$$
Now there's also a subscript that says that the dimention of $N(A)$ is $r$ and the dimension of $C(A^T)$ is $n - r$, but why?
And I suppose that $n$ is the number of columns of $A$, since I have the following $A : R^n \rightarrow R^m$, and I am not seeing the connection between these dimensions...

A:

The connection can easily be seen by Gaussian elimination, i.e. reduction to echelon form.  The number of leading $1$'s in the reduced matrix is the dimension of the row space of $A$, which corresponds to the column space of $A^T$.  The number of columns that do not have leading $1$'s in the reduced matrix is the dimension of the null space of $A$.  The sum of these two dimensions is the total number of columns, which is $n$.

