Page 205 - Matrix Analysis & Applied Linear Algebra

200 Chapter 4 Vector Spaces

In loose terms, this is a kind of conservation law—it says that as the amount
of “stuff” in R (A) increases, the amount of “stuff” in N (A) must decrease,
and vice versa. The phrase rank plus nullity is used because dim R (A) is the
rank of A, and dim N (A) was traditionally known as the nullity of A.
Example 4.4.4

Problem: Determine the dimension as well as a basis for the space spanned by

     
1 1 5
 
2
S =       .
,
6
0
,
1 2 7
 
Solution 1: Place the vectors as columns in a matrix A, and reduce
   
115 103
A =  206  −→ E A =  012  .
127 000
Since span (S)= R (A), we have

dim span (S) = dim R (A)= rank (A)=2.

1 1

The basic columns B = 2 , 0 are a basis for R (A)= span (S) .
1 2
Other bases are also possible. Examining E A reveals that any two vectors in S
form an independent set, and therefore any pair of vectors from S constitutes
a basis for span (S) .

Solution 2: Place the vectors from S as rows in a matrix B, and reduce B
to row echelon form:

   
121 1 2 1
B =  102  −→ U =  0 −21  .
567 0 0 0

T
This time we have span (S)= R B , so that
T
dim span (S) = dim R B = rank (B)= rank (U)=2,

and a basis for span (S)= R B T is given by the nonzero rows in U.