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1.抽象代數導論 (Introduction to Abstract Algebra)
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2.張量分析 (Tensor Analysis)
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3.正交函數展開 (Orthogonal Function Expansion)
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4.格林函數 (Green's Function)
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5.變分法 (Calculus of ariation)
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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
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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. *##+
$+ :off2an. @+. @une. +. Linear Algebra. &nd ed. The 1outheast ,oo- Co. "e3 >ersey. *#5*+
5+ /cCoy. "+ :+. Fundamentals of Abstract Algebra. expanded ?ersion. Allyn B ,acon Inc. ,oston. *#5&+
%+ :ildebrand. F+ ,+. Methods of Applied Mathematics. &nd ed. !rentice9:all Inc. "e3 >ersey. *#5&++
#+ ,urton. 8+ /+. An ntroduction to Abstract Mathematical Systems. Addison9
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8a?id :ilbert8a?id :ilbert BornBorn >anuary &7. *%$& anuary &7. *%$&
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!hilip /+ /orse!hilip /+ /orse
HHOperationsOperations research is anresearch is anapplied science utiliing all -no3napplied science utiliing all -no3n
scientific techniues as tools inscientific techniues as tools in
sol?ing a specific proble2+Jsol?ing a specific proble2+J
Founding O1A !resident (*#6&)Founding O1A !resident (*#6&)
,+1+ !hysics. *#&$. 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
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Francis ,+ :ildebrandFrancis ,+ :ildebrand
George Arf-enGeorge Arf-en
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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
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/01-
34 -
抽象代數56
( ) ( ) ( ) ( ) RV R R R ,,,,,,, •+•+78
eaaaa =+=+ −− 11
( )•, R
性之分佈對、 +•
( ) ( ) ( )
( ) ( ) ( )acabacb
cabacba
•+•=•+•+•=+•
349
:;
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群胚 Groupoid
M A goupoid 2ust satisfy
is closed under the rule of co2bination R
( )+, R
R baR b,a ∈+⇒∈∀
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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
&
*
∗
∗ ∗ ∗∗ ∗∗ ∗
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M A se2igroup is a groupoid 3hose
operation satisfies the associati?e la3+
(groupoid)
半群 1e2igroup
( ) ( ) c bac baR c, b,a ++=++⇒∈∀
R baR b,a ∈+⇒∈∀
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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
+++=++=
+++=+++=+++
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M A se2igroup ha?ing an identity ele2entfor the operation is called a 2onoid+
(groupoid) (se2igroup)
單 /onoid
( )+, R+
aaeeaR a =+=+⇒∈∃∈∀ Re
e
R baR b,a ∈+⇒∈∀
( ) ( ) c bac baR c, b,a ++=++⇒∈∀
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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 ⊆
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群 Group
M A 2onoid 3hich each ele2ent of has
an in?erse is called a group
(groupoid)
(se2igroup)
(2onoid)
( )+, R R
R baR b,a ∈+⇒∈∀
( ) ( ) c bac baR c, b,a ++=++⇒∈∀
aaeeaR a =+=+⇒∈∃∈∀ ReeaaaaR aR a 1-1-1 =+=+⇒∈∃∈∀ −
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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 =+++
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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
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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,•∴
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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,
=+=+⇒∈∃∈∀
=+=+⇒∈∃∈∀++=++⇒∈∀
∈+⇒∈∀+=+⇒∈∀
−
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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
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/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, •+
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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
=•=•⇒∈∃⇒
•+•=•+∧•+•=+•⇒
••=••⇒∈•⇒
=+=+⇒∈∃ =+=+⇒∈∃
++=++⇒∈+⇒
+=+⇒∈∀
−
,
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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-
∈•
∧=+=+⇒∈∃
∧=+=+⇒∈∃
∧++=++
∧∈+
∧+=+⇒∈∀
∧⊆
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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
∈+++=+•+
∈+=++
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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
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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 ∈
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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, +
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&+ 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, •+
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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
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nmi/i/
nmi/
nmnm
0 caac
0 a Rc
nm
0 0
×
×
××
∈=
∈∈
+× +
bytion2ultiplicascalar define.andFor
addition+2atrixof operationtheisand2atricesallof
settheis3here.begroupeco22utati?the0et
)()(
)(
),(
*
Q ector 1pace
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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
∈∈∈∈+∈
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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 /
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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 ∈
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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
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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
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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
-
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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
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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()
-
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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
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Eigen?alues and Eigen?ectors
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8iagonaliation >?@
A suare 2atrix is said to be a diagonal 2atrix if all of its entries are
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A xxyx.
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Canonical For2 EFCD
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Ex 0et T be the linear operator on 7 3hich is represented in
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Ex 0et T be the linear operator on 7 3hich is represented in
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