Say there is some 4 x 5 matrix A. Its reduced row echelon form has three "leading 1's" and has a column of zeroes as well as a row of zeroes.
The rank of the matrix, as I have understand it, would be 3 for this particular matrix as that is the dimension of the row space and the dimension of the column space.
However, I am unsure of what the nullity of this matrix should be. According to my textbook, the nullity of some m x n matrix would be the number of parameters in the general solution Ax = 0. As it turns out, there is only one parameter in the general solution of A. So by the textbook's definition, the nullity of A would be 1.
But my professor has also mentioned the rank-nullity theorem, and that for any m x n matrix A the following equation holds true:
$$rank(A) + nullity(A) = n$$
where n is the number of columns in the matrix. If I am to acknowledge the column of zeroes in the 4x5 matrix, then nullity(A) = 5 - 3 = 2. This is a different result from that of the textbook's definition.
I believe I am misunderstanding something here about how I should regard columns of zeroes in the matrix. Do I account for them in determining the nullity of my matrix? Or are they to be ignored when calculating nullity? My intuition is that I should ignore them, but my intuition has often failed me in mathematics.