chapter1 algebra (1)

8/16/2019 Chapter1 Algebra (1)

1/46

1.抽象代數導論 (Introduction to Abstract Algebra)

-

2.張量分析 (Tensor Analysis)

-

3.正交函數展開 (Orthogonal Function Expansion)

-

4.格林函數 (Green's Function)

-

5.變分法 (Calculus of ariation)

-

6.攝動理論 (!erturbation Theory)

"#$%& 高等工程數學 ※ 先修課程：微積分工程數學 ( 一 )-( 三


2/46

eference*+ ,ir-hoff. G+. /ac0ane. 1+. A Survey of Modern Algebra. &nd ed. The /ac2illan Co. "e3 4or-. *#56+

2 . 徐誠浩 . 抽象代數 - 方法導引 . 復旦大學 . *#%#+

7+ Arangno. 8+ C+. Schaum’s Outline of Theory and Problems of Abstract Algebra. /cGra39:ill Inc. *###+

;+ 8es-ins. o?ano?ich Inc. *##+

$+ :off2an. @+. @une. +. Linear Algebra. &nd ed. The 1outheast ,oo- Co. "e3 >ersey. *#5*+

5+ /cCoy. "+ :+. Fundamentals of Abstract Algebra. expanded ?ersion. Allyn B ,acon Inc. ,oston. *#5&+

%+ :ildebrand. F+ ,+. Methods of Applied Mathematics. &nd ed. !rentice9:all Inc. "e3 >ersey. *#5&++

#+ ,urton. 8+ /+. An ntroduction to Abstract Mathematical Systems. Addison9


3/46

8a?id :ilbert8a?id :ilbert BornBorn >anuary &7. *%$& anuary &7. *%$&


4/46

!hilip /+ /orse!hilip /+ /orse

HHOperationsOperations research is anresearch is anapplied science utiliing all -no3napplied science utiliing all -no3n

scientific techniues as tools inscientific techniues as tools in

sol?ing a specific proble2+Jsol?ing a specific proble2+J

Founding O1A !resident (*#6&)Founding O1A !resident (*#6&)

,+1+ !hysics. *#&$. Case InstituteK,+1+ !hysics. *#&$. Case InstituteK

!h+8+ !hysics. *#. !rinceton!h+8+ !hysics. *#. !rinceton

Dni?ersity+Dni?ersity+

Faculty 2e2ber at /IT. *#7*9*#$#+Faculty 2e2ber at /IT. *#7*9*#$#+

/ethods of Operations esearch/ethods of Operations esearch

Lueues. In?entories. and /aintenanceLueues. In?entories. and /aintenance

0ibrary Effecti?eness0ibrary Effecti?eness

Luantu2 /echanicsLuantu2 /echanics

/ethods of Theoretical !hysics/ethods of Theoretical !hysics

ibration and 1oundibration and 1ound

Theoretical AcousticsTheoretical Acoustics

Ther2al !hysicsTher2al !hysics

:andboo- of /athe2atical Functions. 3ith For2ulas.:andboo- of /athe2atical Functions. 3ith For2ulas.

Graphs. and /athe2atical TablesGraphs. and /athe2atical Tables


5/46

Francis ,+ :ildebrandFrancis ,+ :ildebrand

George Arf-enGeorge Arf-en


6/46

Introduction to Abstract AlgebraIntroduction to Abstract Algebra

抽象代數導論抽象代數導論

M !reli2inary notions!reli2inary notionsM 1yste2s 3ith a single operation1yste2s 3ith a single operationM /athe2atical syste2s 3ith t3o operations/athe2atical syste2s 3ith t3o operationsM /atrix theory an algebraic ?ie3/atrix theory an algebraic ?ie3


7/46

-

/01-

34 -

抽象代數56

( ) ( ) ( ) ( ) RV R R R ,,,,,,, •+•+78

eaaaa =+=+ −− 11

( )•, R

性之分佈對、 +•

( ) ( ) ( )

( ) ( ) ( )acabacb

cabacba

•+•=•+•+•=+•

349

:;


8/46

群胚 Groupoid

M A goupoid 2ust satisfy

is closed under the rule of co2bination R

( )+, R

R baR b,a ∈+⇒∈∀


9/46

M Ex+ Consider the operation defined on the set 1N

*.&.7P by the operation table belo3+

Fro2 the table. 3e see

& (* 7)N& 7N& but (& *) 7N7 7N*The associati?e la3 fails to hold in this groupoid(1. )

&

*

7

* * & 7

*

&

7

*

7

&

7

&

*

∗

∗ ∗ ∗∗ ∗∗ ∗


10/46

M A se2igroup is a groupoid 3hose

operation satisfies the associati?e la3+

(groupoid)

半群 1e2igroup

( ) ( ) c bac baR c, b,a ++=++⇒∈∀



11/46

M Ex+ If the operation is defined on by a b N 2ax a.b P.that is a b is the larger of the ele2ents a and b. or

either one if aNb+

a (b c) N 2ax a. b. c P N (a b) c

that sho3s to be a se2igroup

M If and is a se2igroup. then

proof.

) ,(R# ∗

∗∗

∗ ∗ ∗ ∗

# R ∗

Rd c,b,a, ∈ )(R,+d)c)((bad)(cb)(a +++=+++

d)(cb)(a

xb)(a

xbyd)(c denoted x)(ba

d))(c(bad)c)((ba

+++=++=

+++=+++=+++


12/46

M A se2igroup ha?ing an identity ele2entfor the operation is called a 2onoid+

(groupoid) (se2igroup)

單 /onoid

( )+, R+

aaeeaR a =+=+⇒∈∃∈∀ Re

e


( ) ( ) c bac baR c, b,a ++=++⇒∈∀


13/46

Ex+ ,oth the se2igroups and are instances of2onoids

for each

The e2pty set is the identity ele2ent for the unionoperation+

for each

The uni?ersal set is the identity ele2ent for the

intersection operation+

) ,(S U ) ,(S U

A A A =ϕ=ϕ U A ⊆

A AU U A == U A ⊆


14/46

群 Group

M A 2onoid 3hich each ele2ent of has

an in?erse is called a group

(groupoid)

(se2igroup)

(2onoid)

( )+, R R


( ) ( ) c bac baR c, b,a ++=++⇒∈∀

aaeeaR a =+=+⇒∈∃∈∀ ReeaaaaR aR a 1-1-1 =+=+⇒∈∃∈∀ −


15/46

M If is a group and .then

Proof. all 3e need to sho3 is that

fro2 the uniueness of the in?erse of

3e 3ould conclude

a si2ilar argu2ent establishes that

( )+, R Rba, ∈ -1-1-1 abb)(a +=+

eb)(a )a(b )a(bb)(a -1-1-1-1 =+++=+++

ba +-1-1-1 abb)(a +=+

e

aa

)a(ea

)a )b((ba )a(bb)(a

1-

1-

-1-1-1-1

=+=

++=

+++=+++

eb)(a )a(b -1-1 =+++


16/46

Co22utati?e "交#$

group

1-1-1

monoid

semigroup

groupoid

eaaaa Ra Ra

aaeea Re Racb)(ac)(ba Rcb,a,

Rba Rba,

abba Rba,

=+=+⇒∈∃∈∀

=+=+⇒∈∃∈∀ ++=++⇒∈∀

∈+⇒∈∀+=+⇒∈∀

−

Co!!utative

"roupoid Co!!utative

se!i"roup

Co!!utative !onoid

Co!!utative "roup


17/46

M Ex+ consider the set of nu2ber and the

operation of ordinary 2ultiplication. and represents

integer+

*+ Closure

&+ Associate property

7+ Identity ele2ent

;+ Co22utati?e property

is a co22utati?e 2onoid+

Z}ba,!b"aS ∈+=

S !bc)(ad !bd)(ac )!d (c )!b(a Z d c,b,a, ∈+++=+•+⇒∈∀

[ ] [ ] )! (e )!d (c )!b(a )! (e )!d (c )!b(a +•+•+=+•+•+

•

Z # e,d,c,b,a, ∈∀

!$11 +=

)!b(a )!d (c )!d (c )!b(a +•+=+•+

Z d c,b,a, ∈∀

)(S,•∴


18/46

ing %M A ring is a none2pty set 3ith t3o binary

operations and on such that*+ is a co22utati?e group

&+ is a se2igroup

7+ The t3o operations are related by the distributi?e

la3s

) ,(R, •+

semigroup

groupoid

cb)(ac)(ba Rcb,a, Rba Rba,

••=••⇒∈∀∈•⇒∈∀

a)(ca)(bac)(b

c)(ab)(ac)(ba Rcb,a,

•+•=•+

•+•=+•⇒∈∀

R

+ • R )(R,+

)(R,• group

1-1-1

monoid

semigroup

groupoid

eaaaa Ra Ra

aaeea Re Racb)(ac)(ba Rcb,a,

Rba Rba,

abba Rba,

=+=+⇒∈∃∈∀

=+=+⇒∈∃∈∀++=++⇒∈∀

∈+⇒∈∀+=+⇒∈∀

−


19/46

M A ring consists of a none2pty set and t3o

operations. called addition and 2ultiplication and denoted

by and . respecti?ely. satisfying the reuire2ents1. R is closed under addition

2. Commutative

3. Associative

4. Identity element

!. Inverse

". R is closed under multi#lication

$. Associate

%. &istributive la'

) ,(R, •+

+ •

a)(ca)(bac)(bc)(ab)(ac)(ba%&

cb)(ac)(ba'&

Rba&

$(-a)a Ra&

aa$$a R$*&

cb)(ac)(ba+&

abba!&

Rba1& Rcb,a,

group

1-

•+•=•+∧•+•=+•

••=••∈•

=+⇒∈∃=+=+⇒∈∃

++=+++=+

∈+⇒∈∀

R


20/46

/onoid ing 單%

M A 2onoid ring is a ring 3ith identity that is a

se2igroup 3ith identity

monoid

semigroup

groupoid

group

1-1-1

monoid

semigroup

groupoid

aaeea Re

a)(ca)(bac)(b

c)(ab)(ac)(ba

cb)(ac)(ba

Rba

eaaaa Ra

aaeea Re

cb)(ac)(ba

Rba

abba Rcb,a,

=•=•⇒∈∃⇒•+•=•+

∧•+•=+•⇒••=••⇒

∈•⇒

=+=+⇒∈∃

=+=+⇒∈∃

++=++⇒

∈+⇒+=+⇒∈∀

−Ring

Monoid ring

) ,(R, •+


21/46

M ing 3ith co22utati?e property

abb a

a)(ca)(bac)(bc)(ab)(ac)(b a

cb)(ac)(b a

Rb a

eaaaa Ra

aaeea Re

cb)(ac)(b a abb a

Rba Ra,b,c

---

•=•∧•+•=•+∧•+•=+•

∧••=••∧∈•

∧=+=+⇒∈∃

∧=+=+⇒∈∃∧++=++

∧+=+

∧∈+⇒∈∀

111

) ,(R, •+

Co!!utative

Co!!utative !onoid Rin"

aaeea Re

a)(ca)(bac) (b

c)(ab)(ac)(ba

cb)(ac)(ba

Rba

eaaaa Ra

aaeea Re

cb)(ac)(ba

Rba

abba Rb,ca

1-1-1

=•=•⇒∈∃⇒

•+•=•+∧•+•=+•⇒

••=••⇒∈•⇒

=+=+⇒∈∃ =+=+⇒∈∃

++=++⇒∈+⇒

+=+⇒∈∀

−

,


22/46

1ubring %

M The triple is a subring of the ring

*+ is a none2pty subset of

&+ is a subgroup of

7+ is closed under 2ultiplication

) ,(S, •+ ) ,(R, •+

S

)(S,+

S •

)(R,+

R

S ba

eaaaaS a

aaeeaS e

cb)(ac)(ba

S ba

abbaS a,b,c

RS

1-1-1-

∈•

∧=+=+⇒∈∃

∧=+=+⇒∈∃

∧++=++

∧∈+

∧+=+⇒∈∀

∧⊆


23/46

M The 2ini2al set of conditions for deter2ining subrings

0et be ring and Then the triple

is a subring of if and only if

*+ Closed under differences

&+ Closed under 2ultiplication

M Ex+ 0et then is a subring of

. since

This sho3s that is closed under both differences and

products+

RS ∈≠ϕ ) ,(R, •+ ) ,(S, •+ ) ,(R, •+

S ba

S b-aS ba,

∈•∈⇒∈∀

Z}ba,+b"aS ∈+= ) ,(S, •+

numbersrea o a set is R ), , ,(R##

•+integerso# set teis Z Z,d c,b,a, ∈∀

S

S +ad)(bc+bd)(ac )+d (c )+b(a

S +d)-(bc)-(a )+d (c- )+b(a

∈+++=+•+

∈+=++


24/46

Field &M A field is a co22utati?e 2onoid ring in

3hich each nonero ele2ent has an in?erse under

) ,(F, •+

8efinition of Field8efinition of Field

cabacba .

.

.

. .

.

•+•=+•∈•−

+

•+

)(

,

,

,,

.cb.a.ele2entsof tripleeachFor (7)

*Kidentity3ithgroup.eco22utati?ais)((&)

'Kidentity3ithgroup.eco22utati?ais)((*)

thatsuchtion.2ultiplicaandadditioncalled.onandset

none2ptyof consisting)(syste2al2athe2aticaisfield A


25/46

ector '量

M An n9co2ponent. or n9di2ensional. ?ector is an ntuple of real nu2bers 3ritten either in a ro3 or in a

colu2n+

M o3 ?ector

M Colu2n ?ector

called the co2ponents of the ?ectorn is the di2ension of the ?ector

( )n!1 x x x ,,,

n

!

1

x

x

x

#

k R x ∈


26/46

ector space '量()

M A ?ector space( or linear space)o?er the field F consists of the follo3ing

*+ A co22utati?e group 3hose ele2ents are called

?ectors+

V(.) ) ), , ),(.,((V, or •++

group

1-1-1

monoid

semigroup

groupoid

eaaaaV aV a

aaeeaV eV a

cb)(ac)(baV a,b,c

V baV a,b

abbaV a,b

=+=+⇒∈∃∈∀

=+=+⇒∈∃∈∀

++=++⇒∈∀

∈+⇒∈∀+=+⇒∈∀

−

)(V, +


27/46

&+ A field 3hose ele2ents are called scalars+

eaaaa . a

aaeea . e

a)(ca)(bac) (b

c)(ab)(ac)(ba

cb)(ac)(ba . ba

eaaaa . a

aaeea . e

cb)(ac)(ba

. ba

abba . b,ca

1-1-1

1-1-1

=•=•⇒∈∃⇒

=•=•⇒∈∃⇒•+•=•+

∧•+•=+•⇒••=••⇒

∈•⇒

=+=+⇒∈∃

=+=+⇒∈∃++=++⇒

∈+⇒

+=+⇒∈∀

−

−

,

) ,(F, •+


28/46

7+ An operation of scalar 2ultiplication connecting the

group and field 3hich satisfies the properties

∀ ∈ ∈ ∈

+ = +

=

+ = +

=

o

o o o

o o o o

o o o

o

* & * &

* & * &

(a) and . there is defined an ele2ent K

(b) ( ) ( ) ( )K

(c) ( ) ( ).

(d) ( ) ( ) ( )K

(e) * . 3here * is the field identity ele2ent+

c F x V c x V

c c x c x c x

c c x c c x

c x y c x c y

x x

V is closed under left 2ultiplication by scalars


29/46

nmi/i/

nmi/

nmnm

0 caac

0 a Rc

nm

0 0

×

×

××

∈=

∈∈

+× +

bytion2ultiplicascalar define.andFor

addition+2atrixof operationtheisand2atricesallof

settheis3here.begroupeco22utati?the0et

)()(

)(

),(

*

Q ector 1pace


30/46

tion+2ultiplicascalar under closedis(&) of subgroupais(*)

V )+,(),( ++

1ubspace '量()

φ ≠⊆ V ,M 0et () be a ?ector space o?er the field

() is a subspace of ()

The 2ini2u2 conditions that () 2ust satisfy to be a subspace are

.

+,

cx . c x

y x y x

i2plyand

i2plies

∈∈∈∈+∈


31/46

M If () and /() are ?ector spaces o?er the sa2e field. then the

2apping f 0 / is said to be operation-preser2ing if

),()(

),()()(

xc cx

y x y x

=+=+

+and.ele2entsof pair . cV y x ∈∈∀

f preser?es

() and /() are algebraically eui?alent 3hene?er there exists a

one9to9one operation9preser?ing function fro2 onto /


32/46

0inear Transfor2ations +$,#

M 0et and be ?ector spaces+ A linear transformation from into

is a function fro2 the set into 3ith the follo3ing t3oproperties

.),()(

.,),()()(

α α α scalarsand(ii)

(i)

V x x3 x3

V y x y3 x3 y x3

∈∀=

∈∀+=+

()

is function fro2

to

. 5)( V x x3 ∈


33/46

0et and be ?ector spaces o?er the field and let be a

linear transformation fro2 into +

M The null space ("ernal ) of is the set of all ?ectors x in such that (x) 6

-er O S ( ) 'PT x V T x = ∈ =

M If is finite-dimensional . the ran" of is the dimension of the range of

and the nullity of is the dimension of the null space of +

M

M

M

M

M

-er

ran

Th Al b f 0i T f ti


34/46

U x x3 S x3 S infor )),(())(( =

The Algebra of 0inear Transfor2ations

M0et 0 7 and 8 0 be linear transfor2ations. 3ith 7. . and

?ector spaces+

The composition of 8 and

* &

* & * &

* &

and are ?ectors in . then

( )( ) ( ( )) (by definition of )

( ( ) ( )) (by linearity of )

if x x

! T x x ! T x x ! T

! T x T x T

+ = +

= +

o o

* &

* &

( ( )) ( ( )) (by linearity of )

( )( ) ( )( ) (by definition of )

1i2ilarly. 3e ha?e. 3ith in and a scalar.

( )

! T x ! T x !

! T x ! T x ! T

x

! T

α

= +

= +o o o

o ( ) ( ( )) (by definition of )

( ( )) (by linearity of )

( ( )) (by linearity of

x ! T x ! T

! T x T

! T x

α α

α

α

===

o

)

( )( ) (by definition of )

!

! T x ! T α = o o

t ti f 0i T f ti b / t i


35/46

epresentation of 0inear Transfor2ations by /atrices

+$,#-./01

0et be an n9di2ensional ?ector space o?er the field + is a linear

transfor2ation. and α1, α2,…,αn are ordered bases for + If

A

3 3 3 3

aaa3

aaa3

aaa3

n

nn

nnnnnn

nn

nn

),,,(

)9(,),(),(:9,,,:

)(

)(

)(

21

2121

2211

22221122

12211111

α α α

α α α α α α

α α α α

α α α α

α α α α

=

=⇒+++=

+++=+++=

=

nnnn

n

n

aaa

aaa

aaa

A

21

22221

11211

234 A5 0inear Transfor2ation

6 α1, α2,…,αn 7-./

Inner !roductInner !roduct '量89'量89


36/46

Inner !roductInner !roduct '量89'量89

).().((;)

).().().(

).().().((7)

).().((&)if onlyandif ).(

).((*)

scalarsrealareandandin?ectorsare..

+definition

thefro2yi22ediatelfollo3productinner theof propertiesCertain

bydenotedis3hich).(3ritten.andof productinner the

in?ectorst3obeand0et

7

T

abba

cbcacba

cabacba

babaaaa

aa

R cba

ba

bababa

R ba 3

=

+=++=+

= ==

≥

++=

=

⋅⇒++=++=

α α

β α ,

9: 332211

3

2

1

321

321321

bababa

b

b

b

aaa

4 b /bib4 a /aia

It follo3s fro2 the !ythagorean theore2 that the length of the ?ectorIt follo3s fro2 the !ythagorean theore2 that the length of the ?ector


37/46

It follo3s fro2 the !ythagorean theore2 that the length of the ?ectorIt follo3s fro2 the !ythagorean theore2 that the length of the ?ector

θ θ

θ

cos2cos),(

*

),(),(

222

2;12

3

2

2

2

1

2

3

2

2

2

1321

babaab

baba

R ba

aaaaa

aa

a

−+=−⇒=⇒

=⇒++=

++++=

the2bet3eenanglethebeletandin?ectorsnonerobeand0et

bydenotedis?ector theof lengthThe

is

7

aaa

aaa4 a /aia

aa

xx

yy

2

3

2

2

2

1 aaa ++

2

2

2

1 aa +a

**

a!a+

xx

yy

Sb

9a

S

Inner !roduct 1pace '量89()


38/46

Inner !roduct 1pace '量89()

y x x,y y x

V

x y y x

y x y x

y x y x

5 y 5 x 5 y x

5 x y x 5 y x

x x,x

x,x

V x,yV y x

V

nn

nn

T)( and

anconstitutetosaidisproduct.inner its3ithtogether .space?ector The

)((;)

(7)

(&)

if onlyandif )(

)((*)+propertiesfollo3ingthe

hasitif .onproductinner anbetosaidis)(nu2ber realainand

?ectorsof pair e?erytoassignsthatfunctionaspace.?ector realaisIf

=+++=⇒

=

=

===

+=++=+

==≥

β α β α β α

β

β

α

α

β β

α α

2211

11

),(,

),(),(

),(),(

),(),(),(

),(),(),(

spaceproductinner


39/46

Eigen?alues and Eigen?ectors

:;


40/46

rs+eigen?ectothe

of 2ultiplesscalar bydeter2inedoriginthethroughlinesthefixes

operator linear thecase.thisInly+respecti?e.andrseigen?ecto

ingcorrespond3ithTof seigen?alueare&and7thatfollo3sit

and1ince

bydefinedonoperator linear thebe0etEx+ &

3

3 3

x x3 3

=

−=

=

−=

−=

2

1

1

1

,

2

12

2

1

12

14

2

1

1

13

1

1

12

14

1

1

.12

14)(R

** ** && 77

**

&& I # $ %I # $ %

I # %I # % **

&&

77

;; $ I # & %$ I # & %

' I # ' %' I # ' %

TT

8iagonali ation >?@


41/46

8iagonaliation >?@

A suare 2atrix is said to be a diagonal 2atrix if all of its entries are

ero except those on the 2ain diagonal

nλ

λ

λ

2

1

A linear operator on a finite9di2ensional ?ector space is diagonali#able if

there is a basis ?ector for each ?ector of 3hich is an eigenvector of T+

Orthogonaliation of ector 1ets '量-正交@


42/46

∑−

=

−==

−−=⇒−−==

−=⇒===

−=

=⇒=

1

1

23213133221133

2

22

12122

21112121

12

1222

1

1111

s21

)eu,(eu v )u(

u e

uired t=eo


43/46

A xxyx.

2atrixsy22etricais Ay. A xfor2thein3rittenbecaneuationsof setThe

euationstheobtain3e. 3rite3eIf

acallesis

for2theof degreesecondof expressionsho2ogeneou A

3

i/

nnnnnn

nn

nn

i

i

nnnnnnn

A

a

y xa xa xa

y xa xa xa

y xa xa xa

x

A y

orm9uadratic x xa x xa x xa xa xa xa A

=≡

==

=+++

=+++=+++

∂∂

=

+++++++≡ −−

)(

9:

2

1

.222

2211

22222121

11212111

1,131132112

22

222

2

111

Luadratic For2s ABCD

E I

u i ?

a

l e n

t

E I

u i ?

a

l e n

t

/ii/

i /i/

aa

y xa

==

A A x : y x : y

Canonical For2 EFCD


44/46

Canonical For2 EFCD

L AL A'euationthebydefinedis2atrix A'ne3the3here

x' A'x'or x'L ALx'x'L A)x'(L

x'L xeuationthebyx'of ter2sinexpressedbex?ector the0et

T

TTTT

=

===⇒

=

A A

UU8iagonal 2atrix8iagonal 2atrix

If the eigen?alues and corresponding eigen?ectors of the real sy22etric 2atrixIf the eigen?alues and corresponding eigen?ectors of the real sy22etric 2atrix

A are -no3n. a 2atrix L ha?ing this property can be easily constructed A are -no3n. a 2atrix L ha?ing this property can be easily constructed

nnnn ee Aee A λ λ == ,,1111

=

nnnn

n

n

eee

eee

eee

21

22212

12111

L

0et a 2atrix L be constructed in such a 3ay that the ele2ents of the unit ?ectors0et a 2atrix L be constructed in such a 3ay that the ele2ents of the unit ?ectors

ee**. e. e&&.V+.e.V+.enn are the ele2ents of the successi?e colu2ns of Lare the ele2ents of the successi?e colu2ns of L

eigen?alueeigen?alue

eigen?ector eigen?ector

=

=

nn

n

n

n

n e

e

e

e

e

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e

e

2

1

1

12

11

1


45/46

/atrixOrthogonal ILL or LL

L AL resulttheobtain3eLbyabo?eeuationthe

of 2e2berseualtheyingpre2ultiplbyandexists.Lin?ersetheThus

'SLSthatfollo3sitt.independenlinearlyaree.+++++.e?ectorsthe1ince

LL Aor L A

T*9T

*9*9

*9

n*

⇒==⇒

=

≠

⋅=

=⇒

9:

2

1

2211

2222121

1212111

i/i

nnnnnn

nn

nn

eee

eee

eee

δ λ

λ

λ

λ

λ λ λ

λ λ λ

λ λ λ

iiiii ee x λ λ ==⇒ ∑ A A 2

Ex 0et T be the linear operator on 7 3hich is represented in


46/46

Ex 0et T be the linear operator on 7 3hich is represented in

the standard ordered basis by the 2atrix A

; A

chapter1 algebra (1)

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