MA 35100 LECTURE NOTES: FINAL EXAM REVIEW
§
4.1: Introduction to Linear Spaces (cont’d)
Summary 4.1.6
To find a basis for a linear space
V
, perform the following:
#1.
Choose an arbitrary element
f
∈
V
.
#2.
Find a list of elements
f
1
, f
2
, . . . , f
n
∈
V
and constants
c
1
, c
2
, . . . , c
n
such that
f
=
c
1
f
1
+
c
2
f
2
+
· · ·
+
c
n
f
n
. Then
B
= (
f
1
, f
2
, . . . , f
n
)
spans
V
.
#3.
Verify that the elements
f
1
, f
2
, . . . , f
n
are linearly independent i.e., find all solutions to the
equation
c
1
f
1
+
c
2
f
2
+
· · ·
+
c
n
f
n
= 0
. Then
B
= (
f
1
, f
2
, . . . , f
n
)
is a basis for
V
.
Definition 4.1.8
A linear space
V
is called
finite dimensional
if it has a finite basis
B
=
(
f
1
, f
2
, . . . , f
n
)
, so that we can define dim
(
V
) =
n
. Otherwise, the space is called
infinite dimen
sional.
§
4.2: Linear Transformations and Isomorphisms
Definition 4.2.1
Consider two linear spaces
V
and
W
.
a.
A function
T
:
V
→
W
is called a
linear transformation
if it preserves linear combi
nations. Explicitly, for all
f, g
∈
V
and scalars
k
,
i.
T
(
f
+
g
) =
T
(
f
) +
T
(
g
)
.
ii.
T
(
k f
) =
k T
(
g
)
.
b.
The
kernel
of
T
is ker
(
T
) =
{
f
∈
V
:
T
(
f
) = 0
}
. It is a subset of the domain
V
.
c.
The
image
of
T
is im
(
T
) =
{
g
∈
W
:
g
=
T
(
f
)
for some
f
∈
V
}
.
It is a subset of the
codomain
W
.
Theorem.
Let
T
:
V
→
W
be a linear transformation. Then ker
(
T
)
is a subspace of
V
, and
im
(
T
)
is a subspace of
W
.
Definition.
Let
T
:
V
→
W
be a linear transformation. The
rank
of
T
is dim
(
im
(
T
))
, and
the
nullity
of
T
is dim
(
ker
(
T
))
.
Theorem.
Let
T
:
V
→
W
be a linear transformation between two finite dimensional vector
spaces. Then rank
(
T
) +
null
(
T
) =
dim
(
V
)
.
Definition 4.2.2
Let
T
:
V
→
W
be a linear transformation. We say that
T
is an
isomorphism
if
T
is invertible.
Theorem 4.2.3
Let
V
be a finite dimensional linear space with basis
B
= (
f
1
, f
2
, . . . , f
n
)
. The
B
coordinate transformation
L
B
:
V
→
R
n
,
f
=
c
1
f
1
+
c
2
f
2
+
· · ·
+
c
n
f
n
7→
[
f
]
B
=
c
1
c
2
.
.
.
c
n
.
is an isomorphism.
Theorem 4.2.4
Let
T
:
V
→
W
be a linear transformation between two finite dimensional
vector spaces. Then the following are equivalent.
i.
T
is an isomorphism.
ii.
ker
(
T
) =
{
0
}
and im
(
T
) =
W
.
iii.
null
(
T
) = 0
and dim
(
V
) =
dim
(
W
)
.
1
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§
4.3: Matrix of a Linear Transformation
Definition 4.3.1
Let
V
be an
n
dimensional linear space with basis
B
. Given a linear trans
formation
T
:
V
→
V
, the unique matrix
B
such that
[
T
(
f
)]
B
=
B
[
f
]
B
for all
f
∈
V
is called the
B
matrix
of
T
.
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 Spring '10
 EGoins
 Linear Algebra, Algebra, linear transformation

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