Fix a matrix
M
of size
n
×
m.
Question.
For what
b
∈
R
n
does the system of equations
Fx
=
b
have a solution?
We can ask this same question in another way. The matrix
M
induces a linear map:
M
:
R
m
→
R
n
.
By deﬁnition the image of
F
is the set of
b
∈
R
n
so that the equation
F
(
x
) =
b
has a solution. This set of all such
b
has structure.
Theorem
1
.
Let
F
be a linear map
R
m
→
R
n
.
1. Let
y
1
,y
2
be elements in the image of
F
. Then
y
1
+
y
2
is also in the image of
F
.
2. Let
y
be an element in the image of
F
and
λ
∈
R
. Then
λy
is also in the image of
F
.
Proof.
Since each of
y
1
,y
2
is in the image of
F
, there are elements
x
1
,x
2
in the domain of
F
, that is, they are elements of
R
m
, so that
F
(
x
1
) =
y
1
, F
(
x
2
) =
y
2
.
Now
F
(
x
1
+
x
2
) = (
our function is linear
)
F
(
x
1
) +
F
(
x
2
) =
y
1
+
y
2
.
This proves the ﬁrst statement.
Since
y
is in the image of
F
, there is an element
x
in the domain of
F
so that
F
(
x
) =
y
.
Then
F
(
λx
) = (
our function is linear
)
λF
(
x
) =
λy.
This says that
λy
is in the image of
F
.
Deﬁnition 1.
Let
S
be a subset of
R
n
.
We say that
S
is a subspace provided
•
If
u,v
are both elements of
S
, then so is
u
+
v.
•
If
u
∈
S
and if
λ
is any real number, then
λ
·
u
∈
S
also.
The above theorem says that the image of a linear map is a subspace. Indeed it is a
fact that any subspace of
R
n
is the image of a linear map. Subspaces can appear in many
other ways besides being images of linear maps.
1
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 Fall '08
 MARKMAN
 Linear Algebra, Algebra, Equations, Vector Space, Linear map

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