Page 422 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

and it follows from Theorem 5.5.5b that

Thus the proof will be complete if we can show that the row space and column space of R have the
same dimension. But the dimension of the row space of R is the number of nonzero rows, and the
dimension of the column space of R is the number of columns that contain leading 1's (Theorem
5.5.6). However, the nonzero rows are precisely the rows in which the leading 1's occur, so the
number of leading 1's and the number of nonzero rows are the same. This shows that the row space
and column space of R have the same dimension.

The dimensions of the row space, column space, and nullspace of a matrix are such important numbers that there is some
notation and terminology associated with them.

DEFINITION

The common dimension of the row space and column space of a matrix A is called the rank of A and is denoted by rank(A);
the dimension of the nullspace of A is called the nullity of A and is denoted by nullity(A).

EXAMPLE 1 Rank and Nullity of a Matrix
Find the rank and nullity of the matrix

Solution

The reduced row-echelon form of A is

(1)

(verify). Since there are two nonzero rows (or, equivalently, two leading 1's), the row space and column space are both

two-dimensional, so rank . To find the nullity of A, we must find the dimension of the solution space of the linear

system . This system can be solved by reducing the augmented matrix to reduced row-echelon form. The resulting

matrix will be identical to 1, except that it will have an additional last column of zeros, and the corresponding system of

equations will be

or, on solving for the leading variables, (2)
It follows that the general solution of the system is