Page 178 - Matrix Analysis & Applied Linear Algebra

4.2 Four Fundamental Subspaces 173

Proof. Statement (4.2.7) is an immediate consequence of (4.2.5). To prove
(4.2.8), suppose that the basic columns in A are in positions b 1 ,b 2 ,...,b r ,
and the nonbasic columns occupypositions n 1 ,n 2 ,...,n t , and let Q 1 be the
permutation matrix that permutes all of the basic columns in A to the left-hand
N m×t ) , where B contains the basic columns and
side so that AQ 1 =( B m×r
N contains the nonbasic columns. Since the nonbasic columns are linear com-
binations of the basic columns—recall (2.2.3)—we can annihilate the nonbasic
columns in N using elementarycolumn operations. In other words, there is a
nonsingular matrix Q 2 such that ( BN ) Q 2 =( B0 ) . Thus Q = Q 1 Q 2 is
a nonsingular matrix such that AQ = AQ 1 Q 2 =( BN ) Q 2 =( B 0 ) , and
col
hence A ∼ ( B 0 ). The conclusion (4.2.8) now follows from (4.2.6).
Example 4.2.3
T
Problem: Determine spanning sets for R (A) and R A , where
 
1223
A =  2413  .
3614
Solution: Reducing A to anyrow echelon form U provides the solution—the
basic columns in A correspond to the pivotal positions in U, and the nonzero
1 2 0 1

rows of U span the row space of A. Using E A = 0 0 1 1 produces
0 0 0 0
   
    1 0
1 2  
   
T
,
1
R (A)= span     and R A = span  2   0  .
2
0
1
  ,  
3 1  
   
1 1
So far, onlytwo of the four fundamental subspaces associated with each
m×n T
matrix A ∈ have been discussed, namely, R (A) and R A . To see
where the other two fundamental subspaces come from, consider again a general
n m
linear function f mapping into , and focus on N(f)= {x | f(x)= 0}
(the set of vectors that are mapped to 0 ). N(f) is called the nullspace of f
(some texts call it the kernel of f), and it’s easyto see that N(f) is a subspace
n
of because the closure properties (A1) and (M1) are satisfied. Indeed, if
x 1 , x 2 ∈N(f), then f(x 1 )= 0 and f(x 2 )= 0, so the linearityof f produces
f(x 1 + x 2 )= f(x 1 )+ f(x 2 )= 0 + 0 = 0 =⇒ x 1 + x 2 ∈N(f). (A1)
Similarly, if α ∈ , and if x ∈N(f), then f(x)= 0 and linearityimplies
f(αx)= αf(x)= α0 = 0 =⇒ αx ∈N(f). (M1)
T
Byconsidering the linear functions f(x)= Ax and g(y)= A y, the
m×n
other two fundamental subspaces defined by A ∈ are obtained. Theyare
n T m
N(f)= {x n×1 | Ax = 0}⊆ and N(g)= y m×1 | A y = 0 ⊆ .