Download Linear Transformations: Properties

A FIRST COURSE
IN LINEAR ALGEBRA
An Open Text by Ken Kuttler
Linear Transformations: Properties
Lecture Notes by Karen Seyffarth∗
Adapted by
LYRYX SERVICE COURSE SOLUTION
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Linear Transformations: Properties
Page 1/9
Recall: Linear Transformations
Definition
A transformation T : Rn → Rm is a linear transformation if it satisfies the
following two properties for all ~x , ~y ∈ Rn and all (scalars) a ∈ R.
1
T (~x + ~y ) = T (~x ) + T (~y )
2
T (a~x ) = aT (~x )
Linear Transformations: Properties
(preservation of addition)
(preservation of scalar multiplication)
Properties
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Composition of Linear Transformations
Definition
Suppose T : Rk → Rn and S : Rn → Rm are linear transformations. The
composite (or composition) of S and T is
S ◦ T : Rk → Rm ,
is defined by
(S ◦ T )(~x ) = S(T (~x )) for all ~x ∈ Rk .
T
Rk
S
Rn
Rm
S ◦T
Be careful with the order of the transformations! We write S ◦ T , but it is
the transformation T that is applied first, followed by the transformation
S.
Linear Transformations: Properties
Composition of Linear Transformations
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Theorem
T
S
Let Rk → Rn → Rm be linear transformations, and suppose that S is
induced by matrix A, and T is induced by matrix B. Then S ◦ T is a linear
transformation, and S ◦ T is induced by the matrix AB.
Linear Transformations: Properties
Composition of Linear Transformations
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Example
Let S : R2 → R2 and T : R2 → R2 be linear transformations defined by
x
x
x
−y
x
S
=
and T
=
for all
∈ R2 .
y
−y
y
x
y
Then S and T are induced by matrices
1
0
0 −1
A=
and B =
,
0 −1
1
0
respectively. The composite of S and T is the transformation
S ◦ T : R2 → R2 defined by
x
x
(S ◦ T )
=S T
,
y
y
and has matrix (or is induced by the matrix)
0 −1
AB =
.
−1
0
Linear Transformations: Properties
Composition of Linear
Transformations
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Example (continued)
Therefore the composite of S and T is the linear transformation
x
x
0 −1
x
−y
(S ◦ T )
= AB
=
=
,
y
y
−1
0
y
−x
for each
x
y
∈ R2 .
Compare this with the composite of T and S which is the linear
transformation
x
y
(T ◦ S)
=
,
y
x
x
for each
∈ R2 .
y
Linear Transformations: Properties
Composition of Linear Transformations
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Inverse of a Linear Transformations
Definition
Suppose T : Rn → Rn and S : Rn → Rn are linear transformations such
that for each ~x ∈ Rn ,
(S ◦ T )(~x ) = ~x and (T ◦ S)(~x ) = ~x .
Then T and S are invertible transformations; S is called an inverse of T ,
and T is called an inverse of S. (Geometrically, S reverses the action of
T , and T reverses the action of S.)
Theorem
Let T : Rn → Rn be a matrix transformation induced by matrix A. Then
A is invertible if and only if T has an inverse. In the case where T has an
inverse, the inverse is unique and is denoted T −1 . Furthermore,
T −1 : Rn → Rn is induced by the matrix A−1 .
Fundamental Identities relating T and T −1
1
T −1 ◦ T = 1Rn
−1
2
Rn
Linear Transformations: Properties
T ◦T
=1
Inverse of a Linear Transformations
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Example
Let T : R2 7→ R2 be a transformation given by
x
x +y
T
=
y
y
1 1
Then T is a linear transformation induced by A =
.
0 1
Notice that the matrix A is invertible. Therefore the transformation T has
an inverse, T −1 , induced by
1 −1
−1
A =
0
1
Linear Transformations: Properties
Inverse of a Linear Transformations
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Example (continued)
Consider the action of T and T −1 .
x
1 1
x
x +y
T
=
=
y
0 1
y
y
x +y
1 −1
x +y
x
T −1
=
=
y
0
1
y
y
Therefore
T
Linear Transformations: Properties
−1
x
x
=
T
y
y
Inverse of a Linear Transformations
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