Page 424 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

But the number of leading variables is the same as the number of leading 1's in the reduced
row-echelon form of A, and this is the rank of A. Thus

The number of free variables is equal to the nullity of A. This is so because the nullity of A is the

dimension of the solution space of , which is the same as the number of parameters in the

general solution [see 3, for example], which is the same as the number of free variables. Thus

The proof of the preceding theorem contains two results that are of importance in their own right.
THEOREM 5.6.4

If A is an matrix, then

(a) rank number of leading variables in the solution of .

(b) nullity number of parameters in the general solution of .

EXAMPLE 2 The Sum of Rank and Nullity
The matrix

has 6 columns, so
This is consistent with Example 1, where we showed that

EXAMPLE 3 Number of Parameters in a General Solution

Find the number of parameters in the general solution of if A is a matrix of rank 3.

Solution

From 4,