11 Matrix Rank
Section III.3 in Chapter Two is principally about the rank of a matrix. It begins by discussing row rank and column rank separately but ultimately shows that the row rank and column rank are the same. The rank of a matrix is defined to be the common value of its row rank and column rank.
Definition. The row space of an $m\times n$ matrix $A$ is defined to be the subspace of $\R^n$ spanned by the rows of $A$. The row rank of $A$ is the dimension of that subspace. The column space of an $m\times n$ matrix $A$ is defined to be the subspace of $\R^m$ spanned by the columns of $A$. The column rank of $A$ is the dimension of that subspace.
It is not hard to see that the row space is not changed by applying a row operation to a matrix. This means that row-equivalent matrices have the same row space and the same row rank. That rank can be determined by putting the matrix into reduced echelon form, which will have the same row rank as the original matrix since it is row-equivalent to the original matrix. The row rank of a reduced echelon form matrix is just the number of non-zero rows in the matrix or, equivalently, the number of leading variables. This is true since the non-zero rows are linearly independent and so form a basis for the row space.
When a row operation is applied to a matrix, the column space can change. However, the column rank remains the same. This means that row-equivalent matrices have the same column rank, and to find the column rank, you can put the matrix into reduced echelon form and check the column rank. But the column rank of a reduced echelon form matrix is just the number of leading variables, since the columns that contain leading entries are linearly independent (they are in fact vectors in the standard basis $\langle\vec e_1,\vec e_2,\dots,\vec e_n\rangle$), and the other columns are linear combinations of the columns that contain leading entries.
Since the number of leading variables in the reduced echelon form of the matrix represents both the row rank and column rank of the original matrix, we see that the row rank and column rank are indeed the same.
If the $m\times n$ matrix $A$ has rank $r$, then the number of leading variables is $r$ and the number of free variables is $n-r$. Note that the number of free variables is the dimension of the space of solutions of the homogeneous system corresponding to the matrix.