Page 204 - Matrix Analysis & Applied Linear Algebra

4.4 Basis and Dimension 199

of an independent set is again independent—see (4.3.7) and (4.3.14)—it follows
that the last m − r rows in P are linearly independent, and hence they con-
T T
stitute a basis for N A . And this implies dim N A = m − r (i.e., the
number of rows in A minus the rank of A). Replacing A by A T shows that
T
dim N A T = dim N (A) is the number of rows in A T minus rank A T .
T
But rank A = rank (A)= r, so dim N (A)= n−r. We deduced dim N (A)
without exhibiting a specific basis, but a basis for N (A) is easy to describe.
Recall that the set H containing the h i ’s appearing in the general solution
(4.2.9) of Ax = 0 spans N (A). Since there are exactly n − r vectors in H,
and since dim N (A)= n − r, H is a minimal spanning set, so, by (4.4.2), H
must be a basis for N (A). Below is a summary of facts uncovered above.

Fundamental Subspaces—Dimension and Bases
For an m × n matrix of real numbers such that rank (A)= r,

• dim R (A)= r, (4.4.7)
• dim N (A)= n − r, (4.4.8)
T
• dim R A = r, (4.4.9)
T
• dim N A = m − r. (4.4.10)
Let P be a nonsingular matrix such that PA = U is in row echelon
form, and let H be the set of h i ’s appearing in the general solution
(4.2.9) of Ax = 0.
• The basic columns of A form a basis for R (A). (4.4.11)
T
• The nonzero rows of U form a basis for R A . (4.4.12)
• The set H is a basis for N (A). (4.4.13)
T
• The last m − r rows of P form a basis for N A . (4.4.14)
For matrices with complex entries, the above statements remain valid
provided that A T is replaced with A .
∗

Statements (4.4.7) and (4.4.8) combine to produce the following theorem.

Rank Plus Nullity Theorem

• dim R (A) + dim N (A)= n for all m × n matrices. (4.4.15)