Page 176 - Matrix Analysis & Applied Linear Algebra

4.2 Four Fundamental Subspaces 171

Column and Row Spaces
m×n
For A ∈ , the following statements are true.
• R (A) = the space spanned by the columns of A (column space).
T
• R A = the space spanned by the rows of A (row space).
• b ∈ R (A) ⇐⇒ b = Ax for some x. (4.2.3)
T T T T
• a ∈ R A ⇐⇒ a = y A for some y . (4.2.4)

Example 4.2.1

T 1 2 3
Problem: Describe R (A) and R A for A = .
2 4 6
Solution: R (A)= span {A ∗1 , A ∗2 , A ∗3 } = {α 1 A ∗1 +α 2 A ∗2 +α 3 A ∗3 | α i ∈ },
but since A ∗2 =2A ∗1 and A ∗3 =3A ∗1 , it’s clear that every linear combination
of A ∗1 , A ∗2 , and A ∗3 reduces to a multiple of A ∗1 , so R (A)= span {A ∗1 } .
2
Geometrically, R (A) is the line in through the origin and the point (1, 2).
T
Similarly, R A = span {A 1∗ , A 2∗ } = {α 1 A 1∗ + α 2 A 2∗ | α 1 ,α 2 ∈ } . But
A 2∗ =2A 1∗ implies that every combination of A 1∗ and A 2∗ reduces to a
T 3
multiple of A 1∗ , so R A = span {A 1∗ } , and this is a line in through
the origin and the point (1, 2, 3).
There are times when it is desirable to know whether or not two matrices
have the same row space or the same range. The following theorem provides the
solution to this problem.

Equal Ranges
For two matrices A and B of the same shape:
T T row
• R A = R B if and only if A ∼ B. (4.2.5)
col
• R (A)= R (B) if and only if A ∼ B. (4.2.6)

row
Proof. To prove (4.2.5), first assume A ∼ B so that there exists a nonsingular
T T
matrix P such that PA = B. To see that R A = R B , use (4.2.4) to
write
T T T T −1 T
a ∈ R A ⇐⇒ a = y A = y P PA for some y
T
T
T
T
⇐⇒ a = z B for z = y P −1
T
⇐⇒ a ∈ R B .