Page 188 - Matrix Analysis & Applied Linear Algebra

4.3 Linear Independence 183

Linear Independence and Matrices
Let A be an m × n matrix.
• Each of the following statements is equivalent to saying that the
columns of A form a linearly independent set.
N (A)= {0}. (4.3.2)
rank (A)= n. (4.3.3)

• Each of the following statements is equivalent to saying that the rows
of A form a linearly independent set.
T
N A = {0}. (4.3.4)
rank (A)= m. (4.3.5)

• When A is a square matrix, each of the following statements is
equivalent to saying that A is nonsingular.
The columns of A form a linearly independent set. (4.3.6)
The rows of A form a linearly independent set. (4.3.7)

Proof. By definition, the columns of A are a linearly independent set when
the only set of α ’s satisfying the homogeneous equation

 
α 1
 α 2 

0 = α 1 A ∗1 + α 2 A ∗2 + ··· + α n A ∗n = A ∗1 | A ∗2 | ··· | A ∗n  . 
.
 . 
α n
is the trivial solution α 1 = α 2 = ··· = α n =0, which is equivalent to saying
N (A)= {0}. The fact that N (A)= {0} is equivalent to rank (A)= n was
demonstrated in (4.2.10). Statements (4.3.4) and (4.3.5) follow by replacing A

by A T in (4.3.2) and (4.3.3) and by using the fact that rank (A)= rank A T .
Statements (4.3.6) and (4.3.7) are simply special cases of (4.3.3) and (4.3.5).
Example 4.3.2
} consisting of distinct unit vectors is a linearly indepen-
Any set {e i 1 , e i 2

,..., e i n
= n. For example, the set of unit vec-
dent set because rank e i 1 | e i 2 |· · · | e i n
1 0 0
 
4
tors {e 1 , e 2 , e 4 } in is linearly independent because rank  0 1 0  =3.
0 0 0
0 0 1