Page 175 - Matrix Analysis & Applied Linear Algebra

170 Chapter 4 Vector Spaces

m×n
In particular, this result means that every matrix A ∈ generates
m
a subspace of by means of the range of the linear function f(x)= Ax.
24 m×n n
Likewise, the transpose of A ∈ defines a subspace of by means
T
of the range of f(y)= A y. These two “range spaces” are two of the four
fundamental subspaces defined by a matrix.
Range Spaces

m×n
The range of a matrix A ∈ is defined to be the subspace
m
R (A)of that is generated by the range of f(x)= Ax. That is,
n
m
R (A)= {Ax | x ∈ }⊆ .
Similarly, the range of A T is the subspace of n defined by

T T m n
R A = {A y | y ∈ }⊆ .
m
Because R (A) is the set of all “images” of vectors x ∈ under
transformation by A, some people call R (A) the image space of A.

The observation (4.2.2) that every matrix–vector product Ax (i.e., every
image) is a linear combination of the columns of A provides a useful character-
ization of the range spaces. Allowing the components of x =(ξ 1 ,ξ 2 ,...,ξ n ) T to
vary freely and writing

 
ξ 1
n

 ξ 2 
Ax = A ∗1 | A ∗2 | ··· | A ∗n  .  = ξ j A ∗j
. j=1
 . 
ξ n
shows that the set of all images Ax is the same as the set of all linear combi-
nations of the columns of A. Therefore, R (A) is nothing more than the space
spanned by the columns of A. That’s why R (A) is often called the column
space of A.
T T
Likewise, R A is the space spanned by the columns of A . But the

columns of A T are just the rows of A (stacked upright), so R A T is simply
25 T
the space spanned by the rows of A. Consequently, R A is also known as
the row space of A. Below is a summary.
24
For ease of exposition, the discussion in this section is in terms of real matrices and real spaces,
but all results have complex analogs obtained by replacing A T by A .
∗
25
Strictly speaking, the range of A T is a set of columns, while the row space of A is a set of
rows. However, no logical difficulties are encountered by considering them to be the same.