Advanced Calculus (II)

Advanced Calculus (II)

W^EN-C^HINGL^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 :Rⁿ → R^m is said to be linear (notation:

T ∈ L(Rⁿ;R^m)) if and only if it satisfies

T (x + y) = T (x) + T (y) and T (αx) = αT (x) for allx, y ∈ Rⁿ and all scalars α.

Notations:x = (x1,x2, . . . ,xn) ∈Rⁿ. [x] = [x1x2 · · · x_n]

WEN-CHINGLIEN Advanced Calculus (II)

Remark (8.13)

Ifx, y ∈ Rⁿ 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 lete1, . . . ,en represent the usual basis of Rⁿ. If

(6) T (x) = Bx, x ∈ Rⁿ,

then T is a linear function fromRⁿtoR^m and

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

WEN-CHINGLIEN Advanced Calculus (II)

Theorem (8.15)

For each T ∈ L(Rⁿ;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(Rⁿ;R^m). Define B by (7). Then

T (x) = T

n

X

j=1

xjej

!

=

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(Rⁿ;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(Rⁿ;R^m).Define B by (7). Then

T (x) = T

n

X

j=1

xjej

!

=

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(Rⁿ;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(Rⁿ;R^m). Define B by (7).Then

T (x) = T

n

X

j=1

xjej

!

=

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(Rⁿ;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(Rⁿ;R^m).Define B by (7). Then

T (x) = T

n

X

j=1

xjej

!

=

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(Rⁿ;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(Rⁿ;R^m). Define B by (7).Then

T (x) = T

n

X

j=1

xjej

!

=

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(Rⁿ;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(Rⁿ;R^m). Define B by (7). Then

T (x) = T

n

X

j=1

xjej

!

=

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(Rⁿ;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(Rⁿ;R^m). Define B by (7). Then

T (x) = T

n

X

j=1

xjej

!

=

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(Rⁿ;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(Rⁿ;R^m). Define B by (7). Then

T (x) = T

n

X

j=1

xjej

!

=

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(Rⁿ;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(Rⁿ;R^m). Define B by (7). Then

T (x) = T

n

X

j=1

xjej

!

=

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(Rⁿ;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 ∈ Rⁿ} 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(Rⁿ;R^m). Then the operator norm of T is finite, and satisfies

(8) kT (x)k ≤ kT k kxk

for allx ∈ Rⁿ. Proof.

Let B be the m × n matrix that represents T , and suppose that the rows of T are given byb1, . . . ,bm.By the

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

T (x) = (b1· x, . . . , bm· x).

WEN-CHINGLIEN Advanced Calculus (II)

Theorem (8.17)

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

(8) kT (x)k ≤ kT k kxk

for allx ∈ Rⁿ. Proof.

Let B be the m × n matrix that represents T , and suppose that the rows of T are given byb1, . . . ,bm.By the

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

T (x) = (b1· x, . . . , bm· x).

WEN-CHINGLIEN Advanced Calculus (II)

Theorem (8.17)

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

(8) kT (x)k ≤ kT k kxk

for allx ∈ Rⁿ. Proof.

Let B be the m × n matrix that represents T , and suppose that the rows of T are given byb1, . . . ,bm.By the

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

T (x) = (b1· x, . . . , bm· 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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk²

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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk²

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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk²

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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk²

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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk²

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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk² 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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk² 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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk² 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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk² 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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk² 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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk² 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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk² 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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk² 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 ≤ Ckkxk for all x ∈ Rⁿ.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)k² = (b1· x)²+ . . . + (bm· x)²

≤ (kb1k kxk)²+ . . . + (kbmk kxk)²

≤ m · max{kb_jk²:1 ≤ j ≤ m} kxk² =:C kxk² 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 ≤ Ckkxk for all x ∈ Rⁿ.Taking the limit of this last inequality as k → ∞, we obtain (8).

WEN-CHINGLIEN Advanced Calculus (II)

Thank you.

WEN-CHINGLIEN Advanced Calculus (II)