Page 421 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

5.6 In the preceding section we investigated the relationships between systems
of linear equations and the row space, column space, and nullspace of the
RANK AND NULLITY coefficient matrix. In this section we shall be concerned with relationships
between the dimensions of the row space, column space, and nullspace of a
matrix and its transpose. The results we will obtain are fundamental and will
provide deeper insights into linear systems and linear transformations.

Four Fundamental Matrix Spaces

If we consider a matrix A and its transpose together, then there are six vector spaces of interest:

row space of A row space of

column space of A column space of

nullspace of A nullspace of

However, transposing a matrix converts row vectors into column vectors and column vectors into row vectors, so except for a
difference in notation, the row space of is the same as the column space of A, and the column space of is the same as the
row space of A. This leaves four vector spaces of interest:

row space of A column space of A

nullspace of A nullspace of

These are known as the fundamental matrix spaces associated with A. If A is an matrix, then the row space of A and the

nullspace of A are subspaces of , and the column space of A and the nullspace of are subspaces of . Our primary goal

in this section is to establish relationships between the dimensions of these four vector spaces.

Row and Column Spaces Have Equal Dimensions

In Example 6 of Section 5.5, we found that the row and column spaces of the matrix

each have three basis vectors; that is, both are three-dimensional. It is not accidental that these dimensions are the same; it is a
consequence of the following general result.
THEOREM 5.6.1

If A is any matrix, then the row space and column space of A have the same dimension.

Proof Let R be any row-echelon form of A. It follows from Theorem 5.5.4 that