Page 179 - Matrix Analysis & Applied Linear Algebra

174 Chapter 4 Vector Spaces

Nullspace

n
• For an m × n matrix A, the set N (A)= {x n×1 | Ax = 0}⊆
is called the nullspace of A. In other words, N (A) is simply the
set of all solutions to the homogeneous system Ax = 0.
T T m
• The set N A = y m×1 | A y = 0 ⊆ is called the left-

hand nullspace of A because N A T is the set of all solutions
T
T
to the left-hand homogeneous system y A = 0 .

Example 4.2.4
1 2 3
Problem: Determine a spanning set for N (A), where A = .
2 4 6
Solution: N (A) is merely the general solution of Ax = 0, and this is deter-
mined by reducing A to a row echelon form U. As discussed in §2.4, any such

1 2 3
U will suffice, so we will use E A = . Consequently, x 1 = −2x 2 −3x 3 ,
0 0 0
where x 2 and x 3 are free, so the general solution of Ax = 0 is
       
−2 −3
x 1 −2x 2 − 3x 3
= 1 0 .
x 2 x 2 = x 2 + x 3
       
x 3 x 3 0 1
In other words, N (A) is the set of all possible linear combinations of the vectors

   
−2 −3
h 1 =  1  and h 2 =  0  ,
0 1

and therefore span {h 1 , h 2 } = N (A). For this example, N (A) is the plane in
3
that passes through the origin and the two points h 1 and h 2 .

Example 4.2.4 indicates the general technique for determining a spanning
set for N (A). Below is a formal statement of this procedure.