Page 423 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

or, equivalently,

(3)

Because the four vectors on the right side of 3 form a basis for the solution space, nullity .
The following theorem states that a matrix and its transpose have the same rank.
THEOREM 5.6.2

If A is any matrix, then .

Proof

The following theorem establishes an important relationship between the rank and nullity of a matrix.
THEOREM 5.6.3

Dimension Theorem for Matrices
If A is a matrix with n columns, then

(4)

Proof Since A has n columns, the homogeneous linear system has n unknowns (variables). These fall into two
categories: the leading variables and the free variables. Thus