## Advanced Calculus (II)

W^{EN}-C^{HING}L^{IEN}

Department of Mathematics National Cheng Kung University

2009

WEN-CHINGLIEN **Advanced Calculus (II)**

## Ch8: Euclidean Spaces

### 8.2: Planes and Linear Transformations

Definition (8.12)

A function T :**R**^{n} **→ R**^{m} is said to be linear (notation:

T ∈ L(R^{n};**R**^{m})) if and only if it satisfies

T (x + y) = T (x) + T (y) and T (αx) = αT (x)
for all**x, y ∈ R**^{n} and all scalars α.

Notations:**x = (x**1,x2, . . . ,xn) ∈**R**^{n}.
[x] = [x1x2 · · · x_{n}]

WEN-CHINGLIEN **Advanced Calculus (II)**

Remark (8.13)

If**x, y ∈ R**^{n} and α is a scalar, then

[x + y] = [x] + [y], [x · y] = [x][y]^{T}, and [αx] = α[x].

WEN-CHINGLIEN **Advanced Calculus (II)**

Remark (8.14)

Let B = [bij]be an m × n matrix whose entries and real
numbers, and let**e****1**, . . . ,**e****n** represent the usual basis of
**R**^{n}. If

(6) T (x) = Bx, **x ∈ R**^{n},

then T is a linear function from**R**^{n}to**R**^{m} and

(7) T (e**j**) = (b1j,b2j, . . . ,bmj), j = 1, 2, . . . , n.

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (8.15)

For each T ∈ L(R^{n};**R**^{m})there is a matrix B = [bij]m×n such
that (6) holds. Moreover, the matrix B is unique.

Specifically, for each fixed T there is only one B that satisfies (6), and the entries of that B are defined by (7).

Proof.

Uniqueness has been established in Remark 8.14. To
prove existence,suppose that T ∈ L(R^{n};**R**^{m}). Define B by
(7). Then

T (x) = T

n

X

j=1

xj**e**j

!

=

n

X

j=1

xjT (ej) =

n

X

j=1

xj(b1j,b2j, . . . ,bmj)

=

n

X

j=1

xjb1j,

n

X

j=1

xjb2j, . . . ,

n

X

j=1

xjbmj

!

=Bx.

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (8.15)

For each T ∈ L(R^{n};**R**^{m})there is a matrix B = [bij]m×n such
that (6) holds. Moreover, the matrix B is unique.

Specifically, for each fixed T there is only one B that satisfies (6), and the entries of that B are defined by (7).

Proof.

Uniqueness has been established in Remark 8.14.To
prove existence, suppose that T ∈ L(R^{n};**R**^{m}).Define B by
(7). Then

T (x) = T

n

X

j=1

xj**e**j

!

=

n

X

j=1

xjT (ej) =

n

X

j=1

xj(b1j,b2j, . . . ,bmj)

=

n

X

j=1

xjb1j,

n

X

j=1

xjb2j, . . . ,

n

X

j=1

xjbmj

!

=Bx.

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (8.15)

For each T ∈ L(R^{n};**R**^{m})there is a matrix B = [bij]m×n such
that (6) holds. Moreover, the matrix B is unique.

Specifically, for each fixed T there is only one B that satisfies (6), and the entries of that B are defined by (7).

Proof.

Uniqueness has been established in Remark 8.14. To
prove existence,suppose that T ∈ L(R^{n};**R**^{m}). Define B by
(7).Then

T (x) = T

n

X

j=1

xj**e**j

!

=

n

X

j=1

xjT (ej) =

n

X

j=1

xj(b1j,b2j, . . . ,bmj)

=

n

X

j=1

xjb1j,

n

X

j=1

xjb2j, . . . ,

n

X

j=1

xjbmj

!

=Bx.

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (8.15)

^{n};**R**^{m})there is a matrix B = [bij]m×n such
that (6) holds. Moreover, the matrix B is unique.

Proof.

Uniqueness has been established in Remark 8.14. To
prove existence, suppose that T ∈ L(R^{n};**R**^{m}).Define B by
(7). Then

T (x) = T

n

X

j=1

xj**e**j

!

=

n

X

j=1

xjT (ej) =

n

X

j=1

xj(b1j,b2j, . . . ,bmj)

=

n

X

j=1

xjb1j,

n

X

j=1

xjb2j, . . . ,

n

X

j=1

xjbmj

!

=Bx.

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (8.15)

^{n};**R**^{m})there is a matrix B = [bij]m×n such
that (6) holds. Moreover, the matrix B is unique.

Proof.

Uniqueness has been established in Remark 8.14. To
prove existence, suppose that T ∈ L(R^{n};**R**^{m}). Define B by
(7).Then

T (x) = T

n

X

j=1

xj**e**j

!

=

n

X

j=1

xjT (ej)=

n

X

j=1

xj(b1j,b2j, . . . ,bmj)

=

n

X

j=1

xjb1j,

n

X

j=1

xjb2j, . . . ,

n

X

j=1

xjbmj

!

=Bx.

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (8.15)

^{n};**R**^{m})there is a matrix B = [bij]m×n such
that (6) holds. Moreover, the matrix B is unique.

Proof.

Uniqueness has been established in Remark 8.14. To
prove existence, suppose that T ∈ L(R^{n};**R**^{m}). Define B by
(7). Then

T (x) = T

n

X

j=1

xj**e**j

!

=

n

X

j=1

xjT (ej) =

n

X

j=1

xj(b1j,b2j, . . . ,bmj)

=

n

X

j=1

xjb1j,

n

X

j=1

xjb2j, . . . ,

n

X

j=1

xjbmj

!

=Bx.

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (8.15)

^{n};**R**^{m})there is a matrix B = [bij]m×n such
that (6) holds. Moreover, the matrix B is unique.

Proof.

Uniqueness has been established in Remark 8.14. To
prove existence, suppose that T ∈ L(R^{n};**R**^{m}). Define B by
(7). Then

T (x) = T

n

X

j=1

xj**e**j

!

=

n

X

j=1

xjT (ej)=

n

X

j=1

xj(b1j,b2j, . . . ,bmj)

=

n

X

j=1

xjb1j,

n

X

j=1

xjb2j, . . . ,

n

X

j=1

xjbmj

!

=Bx.

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (8.15)

^{n};**R**^{m})there is a matrix B = [bij]m×n such
that (6) holds. Moreover, the matrix B is unique.

Proof.

Uniqueness has been established in Remark 8.14. To
prove existence, suppose that T ∈ L(R^{n};**R**^{m}). Define B by
(7). Then

T (x) = T

n

X

j=1

xj**e**j

!

=

n

X

j=1

xjT (ej) =

n

X

j=1

xj(b1j,b2j, . . . ,bmj)

=

n

X

j=1

xjb1j,

n

X

j=1

xjb2j, . . . ,

n

X

j=1

xjbmj

!

=Bx.

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (8.15)

^{n};**R**^{m})there is a matrix B = [bij]m×n such
that (6) holds. Moreover, the matrix B is unique.

Proof.

^{n};**R**^{m}). Define B by
(7). Then

T (x) = T

n

X

j=1

xj**e**j

!

=

n

X

j=1

xjT (ej) =

n

X

j=1

xj(b1j,b2j, . . . ,bmj)

=

n

X

j=1

xjb1j,

n

X

j=1

xjb2j, . . . ,

n

X

j=1

xjbmj

!

=Bx.

WEN-CHINGLIEN **Advanced Calculus (II)**

Definition (8.16)

Let T ∈ L(R^{n};**R**^{m}). The operator norm of T is the
extended real number

**kT k := inf {C > 0 : kT (x)k ≤ C kxk for all x ∈ R**^{n}}
one interesting corollary of Theorem 8.15 is that the
operator norm of a linear function is always finite.

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (8.17)

Let T ∈ L(R^{n};**R**^{m}). Then the operator norm of T is finite,
and satisfies

(8) **kT (x)k ≤ kT k kxk**

for all**x ∈ R**^{n}.
Proof.

Let B be the m × n matrix that represents T , and suppose
that the rows of T are given by**b**1, . . . ,**b**m.By the

definition of matrix multiplication and our identification of
**R**^{m} with m × 1 matrices,

T (x) = (b**1****· x, . . . , b****m****· x).**

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (8.17)

Let T ∈ L(R^{n};**R**^{m}). Then the operator norm of T is finite,
and satisfies

(8) **kT (x)k ≤ kT k kxk**

for all**x ∈ R**^{n}.
Proof.

Let B be the m × n matrix that represents T , and suppose
that the rows of T are given by**b**1, . . . ,**b**m.By the

definition of matrix multiplication and our identification of
**R**^{m} with m × 1 matrices,

T (x) = (b**1****· x, . . . , b****m****· x).**

WEN-CHINGLIEN **Advanced Calculus (II)**

Theorem (8.17)

Let T ∈ L(R^{n};**R**^{m}). Then the operator norm of T is finite,
and satisfies

(8) **kT (x)k ≤ kT k kxk**

for all**x ∈ R**^{n}.
Proof.

Let B be the m × n matrix that represents T , and suppose
that the rows of T are given by**b**1, . . . ,**b**m.By the

definition of matrix multiplication and our identification of
**R**^{m} with m × 1 matrices,

T (x) = (b**1****· x, . . . , b****m****· x).**

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

If B = O,then kT k = 0 and (8) is an equality.If B 6= O, then by the Cauchy-Schwarz Inequality, the square of the Euclidean norm of T (x) satisfies

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}

and C > 0.Thus the set defining kT k is nonempty. Since it
is bounded below (by 0), it follows from the Completeness
Axiom that kT k exists and is finite.In particular, there are
Ck >0 such that Ck **↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

If B = O, then kT k = 0 and (8) is an equality.If B 6= O, then by the Cauchy-Schwarz Inequality, the square of the Euclidean norm of T (x) satisfies

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}

and C > 0.Thus the set defining kT k is nonempty. Since it
is bounded below (by 0), it follows from the Completeness
Axiom that kT k exists and is finite.In particular, there are
Ck >0 such that Ck **↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

If B = O, then kT k = 0 and (8) is an equality.If B 6= O, then by the Cauchy-Schwarz Inequality, the square of the Euclidean norm of T (x) satisfies

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}

and C > 0.Thus the set defining kT k is nonempty. Since it
is bounded below (by 0), it follows from the Completeness
Axiom that kT k exists and is finite.In particular, there are
Ck >0 such that Ck **↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

If B = O, then kT k = 0 and (8) is an equality.If B 6= O, then by the Cauchy-Schwarz Inequality, the square of the Euclidean norm of T (x) satisfies

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}

**↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}

**↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}
and C > 0.Thus the set defining kT k is nonempty. Since it
is bounded below (by 0), it follows from the Completeness
Axiom that kT k exists and is finite.In particular, there are
Ck >0 such that Ck **↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}
and C > 0.Thus the set defining kT k is nonempty. Since it
is bounded below (by 0), it follows from the Completeness
Axiom that kT k exists and is finite.In particular, there are
Ck >0 such that Ck **↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}
and C > 0.Thus the set defining kT k is nonempty. Since it
is bounded below (by 0), it follows from the Completeness
Axiom that kT k exists and is finite.In particular, there are
Ck >0 such that Ck **↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}
and C > 0.Thus the set defining kT k is nonempty. Since it
is bounded below (by 0),it follows from the Completeness
Axiom that kT k exists and is finite.In particular, there are
Ck >0 such that Ck **↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}
and C > 0.Thus the set defining kT k is nonempty. Since it
is bounded below (by 0), it follows from the Completeness
Axiom that kT k exists and is finite.In particular, there are
Ck >0 such that Ck **↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}
and C > 0.Thus the set defining kT k is nonempty. Since it
is bounded below (by 0),it follows from the Completeness
Axiom that kT k exists and is finite.In particular, there are
Ck >0 such that Ck **↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}
and C > 0.Thus the set defining kT k is nonempty. Since it
is bounded below (by 0), it follows from the Completeness
Axiom that kT k exists and is finite.In particular, there are
Ck >0 such that Ck **↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}
and C > 0.Thus the set defining kT k is nonempty. Since it
is bounded below (by 0), it follows from the Completeness
Axiom that kT k exists and is finite.In particular, there are
Ck >0 such that Ck **↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

Proof.

**kT (x)k**^{2} = (b1**· x)**^{2}+ . . . + (bm**· x)**^{2}

**≤ (kb**1**k kxk)**^{2}+ . . . + (kbm**k kxk)**^{2}

**≤ m · max{kb**_{j}k^{2}:1 ≤ j ≤ m} kxk^{2} =:C kxk^{2}
and C > 0.Thus the set defining kT k is nonempty. Since it
is bounded below (by 0), it follows from the Completeness
Axiom that kT k exists and is finite.In particular, there are
Ck >0 such that Ck **↓ kT k and kT (x)k ≤ C**k**kxk for all**
**x ∈ R**^{n}.Taking the limit of this last inequality as k → ∞,
we obtain (8).

WEN-CHINGLIEN **Advanced Calculus (II)**

## Thank you.

WEN-CHINGLIEN **Advanced Calculus (II)**