Page 174 - Matrix Analysis & Applied Linear Algebra

4.2 Four Fundamental Subspaces 169
4.2 FOUR FUNDAMENTAL SUBSPACES

The closure properties (A1) and (M1) on p. 162 that characterize the notion
of a subspace have much the same “feel” as the definition of a linear function as
stated on p. 89, but there’s more to it than just a “similar feel.” Subspaces are
intimately related to linear functions as explained below.

Subspaces and Linear Functions

n m
For a linear function f mapping into , let R(f) denote the
n m
range of f. That is, R(f)= {f(x) | x ∈ }⊆ is the set of all
n
“images” as x varies freely over .
n m
• The range of every linear function f : → is a subspace of
m m
, and every subspace of is the range of some linear function.
m
For this reason, subspaces of are sometimes called linear spaces.
n
Proof. If f : → m is a linear function, then the range of f is a subspace
m
of because the closure properties (A1) and (M1) are satisfied. Establish
(A1) by showing that y 1 , y 2 ∈R(f) ⇒ y 1 +y 2 ∈R(f). If y 1 , y 2 ∈R(f), then
n
there must be vectors x 1 , x 2 ∈ such that y 1 = f(x 1 ) and y 2 = f(x 2 ), so
it follows from the linearity of f that
y 1 + y 2 = f(x 1 )+ f(x 2 )= f(x 1 + x 2 ) ∈R(f).

Similarly, establish (M1) by showing that if y ∈R(f), then αy ∈R(f) for all
scalars α by using the definition of range along with the linearity of f to write

n
y ∈R(f)=⇒ y = f(x) for some x ∈ =⇒ αy = αf(x)= f(αx) ∈R(f).
m
Now prove that every subspace V of is the range of some linear function
n
m
f : → . Suppose that {v 1 , v 2 ,..., v n } is a spanning set for V so that
V = {α 1 v 1 + ··· + α n v n | α i ∈R}. (4.2.1)

Stack the v i ’s as columns in a matrix A m×n = v 1 | v 2 |···| v n , and put the
α i ’s in an n × 1 column x =(α 1 ,α 2 ,...,α n ) T to write

 
α 1
.
.
α 1 v 1 + ··· + α n v n = v 1 | v 2 |···| v n  .  = Ax. (4.2.2)
α n
The function f(x)= Ax is linear (recall Example 3.6.1, p. 106), and we have
n×1
that R(f)= {Ax | x ∈ } = {α 1 v 1 + ··· + α n v n | α i ∈R} = V.