Page 183 - Matrix Analysis & Applied Linear Algebra
P. 183
178 Chapter 4 Vector Spaces
Summary
The four fundamental subspaces associated with A m×n are as follows.
m
• The range or column space: R (A)= {Ax}⊆ .
T T n
• The row space or left-hand range: R A = A y ⊆ .
n
• The nullspace: N (A)= {x | Ax = 0}⊆ .
T T m
• The left-hand nullspace: N A = y | A y = 0 ⊆ .
Let P be a nonsingular matrix such that PA = U, where U is in row
echelon form, and suppose rank (A)= r.
• Spanning set for R (A) = the basic columns in A.
T
• Spanning set for R A = the nonzero rows in U.
• Spanning set for N (A) =the h i ’s in the general solution of Ax = 0.
T
• Spanning set for N A = the last m − r rows of P.
If A and B have the same shape, then
row T T
• A ∼ B ⇐⇒ N (A)= N (B) ⇐⇒ R A = R B .
col T T
• A ∼ B ⇐⇒ R (A)= R (B) ⇐⇒ N A = N B .
Exercises for section 4.2
4.2.1. Determine spanning sets for each of the four fundamental subspaces
associated with
1 2 1 1 5
A = −2 −4 0 4 −2 .
1 2 2 4 9
4.2.2. Consider a linear system of equations A m×n x = b.
(a) Explain why Ax = b is consistent if and only if b ∈ R (A).
(b) Explain why a consistent system Ax = b has a unique solution
if and only if N (A)= {0}.