Page 445 - Schaum's Outline of Theory and Problems of Signals and Systems

432 REVIEW OF MATRIX THEORY [APP. A

EXAMPLE A.8

Thus,

Notes:

1. (A-y =A
2. (A-')~=(A~)-'

Note that if A is invertible, then AB = 0 implies that B = 0 since

A.3 LINEAR INDEPENDENCE AND RANK
A. Linear independence:
Let A=[a, a, ... a,], where ai denotes the ith column vector of A. A set of
column vectors a; (1 = 1,2,. . . , n) is said to be linearly dependent if there exist numbers ai
(i = 1,2,. . . , n) not all zero such that

If Eq. (A.18) holds only for all cui = 0, then the set is said to be linearly independent.

EXAMPLE A.9 Let

Since 2a, + (-3)a, + a, = 0, a,, a,, and a, are linearly dependent. Let

Then

implies that a, = a, = a, = 0. Thus, d,, d,, and d, are linearly independent.