Lect20 - Chapter 4 Linear Transformations Math1111 Linear...

Chapter 4. Linear Transformations Math1111
Linear Transformations Motivation
Recall: A function
f
from a set
V
to set
W
, written as
f : V → W
,
is an assignment of elements in
V
to elements in
W
such that
(i) every element in
V
is sent to exactly one element in
W
,
(ii) every element in
V
is assignment to some element in
W
.
We input an element in
V
into
f
, then get an output in
W

Chapter 4. Linear Transformations Math1111
Linear Transformations Motivation
Recall: A function
f
from a set
V
to set
W
, written as
f : V → W
,
is an assignment of elements in
V
to elements in
W
such that
(i) every element in
V
is sent to exactly one element in
W
,
(ii) every element in
V
is assignment to some element in
W
.
Not allowed

Chapter 4. Linear Transformations Math1111
Linear Transformations Motivation
Recall: A function
f
from a set
V
to set
W
, written as
f : V → W
,
is an assignment of elements in
V
to elements in
W
such that
(i) every element in
V
is sent to exactly one element in
W
,
(ii) every element in
V
is assignment to some element in
W
.
Allowed

Chapter 4. Linear Transformations Math1111
Linear Transformations Motivation
Recall: A function
f
from a set
V
to set
W
, written as
f : V → W
,
is an assignment of elements in
V
to elements in
W
such that
(i) every element in
V
is sent to exactly one element in
W
,
(ii) every element in
V
is assignment to some element in
W
.
Not allowed

Chapter 4. Linear Transformations Math1111
Linear Transformations Motivation
Recall: A function
f
from a set
V
to set
W
, written as
f : V → W
,
is an assignment of elements in
V
to elements in
W
such that
(i) every element in
V
is sent to exactly one element in
W
,
(ii) every element in
V
is assignment to some element in
W
.
Allowed

Chapter 4. Linear Transformations Math1111
Linear Transformations Motivation
Recall: A function
f
from a set
V
to set
W
, written as
f : V → W
,
is an assignment of elements in
V
to elements in
W
such that
(i) every element in
V
is sent to exactly one element in
W
,
(ii) every element in
V
is assignment to some element in
W
.
A vector space is a set endowed with addition and scalar multiplication
(and satisfying some axioms).
Want to study functions preserve these operations.

Chapter 4. Linear Transformations Math1111
Definitions & Examples Linear Transformation - Definition
Definition Let
V
and
W
be vector spaces.
A linear transformation
L : V → W
is a function from
V
into
W
and satisfies
L ( α v
1
+ β v
2
) = α L ( v
1
) + β L ( v
2
)
for all
v
1
, v
2
∈ V
and for all scalars
α
and
β
.

Chapter 4. Linear Transformations Math1111
Definitions & Examples Linear Transformation - Definition
Definition Let
V
and
W
be vector spaces.
A linear transformation
L : V → W
is a function from
V
into
W
and satisfies
L ( α v
1
+ β v
2
) = α L ( v
1
) + β L ( v
2
)
for all
v
1
, v
2
∈ V
and for all scalars
α
and
β
.