Page 184 - Matrix Analysis & Applied Linear Algebra

4.2 Four Fundamental Subspaces 179

4.2.3. Suppose that A is a 3 × 3 matrix such that

     
1 1 −2
   
2
,
R =    −1  and N =  1 
3 2 0
   
span R (A) and N (A), respectively, and consider a linear system
1

Ax = b, where b = −7 .
0
(a) Explain why Ax = b must be consistent.
(b) Explain why Ax = b cannot have a unique solution.
−111 −21 −2
   
 −103 −42   −5 
4.2.4. If A =  −103 −53  and b =  −6  , is b ∈ R (A)?




−103 −64 −7
   
−103 −64 −7
4.2.5. Suppose that A is an n × n matrix.
n
(a) If R (A)= , explain why A must be nonsingular.
(b) If A is nonsingular, describe its four fundamental subspaces.
   
115 1 −44
4.2.6. Consider the matrices A =  206  and B =  4 −86  .
127 0 −45
(a) Do A and B have the same row space?
(b) Do A and B have the same column space?
(c) Do A and B have the same nullspace?
(d) Do A and B have the same left-hand nullspace?

4.2.7. If A = A 1 is a square matrix such that N (A 1 )= R A T , prove
2
A 2
that A must be nonsingular.
T
4.2.8. Consider a linear system of equations Ax = b for which y b =0
T
for every y ∈ N A . Explain why this means the system must be
consistent.
4.2.9. For matrices A m×n and B m×p , prove that

R (A | B)= R (A)+ R (B).