R 3E V = ( f 1 , f 2 ) $<#Q.10G-:#

" #%$'&'(*)%+,#%-/.103254#768(*#79:-;#7<=->2%(*#7<!27#7-@?4BAC->(8DC$!#7-7E

0GFIH'J/KGL!MNL

OQPSRUTWVXRUYCT[Z

•

^\] ?FI<^9:#=_`^#7-[Da$#7-bJc(8FI<-[^(8d

]

9:#7<BJc#%-[->$9e^#%-ef#7$(86*6*#%- ^(8-:Jc(8<2%J>#7- #gJ

<a$+

]

9>F3J

]

#%- 2%FI+h?SFI9bJ50G<BJ[25410327$<#jiIFG-,<FG+ #%Jh?9

]

<FI+ k \

#7<^9:#l03$

+hFI(*<-m$!<#hf#%$(*686*#e?1039nDa$#%-:J>(*FI<poq+hr7+h#[#7<s270G-t^uv0G.-:J>#7<BJc(8FI<xwyk{z|6#7-bJ

03J:Jc#7<!^$}Da$#n6*#%-;9

]

?SFI<!->#7-fFI$9:<(*#%-->FI(8#7<BJ276*0G(*9:#7+h#7<BJ~:$-bJc(8€

]

#7-%k

‚FI<ƒJc9>0WiG03(*6…„

†1‡[ˆŠ‰ ?SFI(8<BJc-:‹…Œ'FG(*#7<BJ

U = ( e 1 , e 2 , e 3 )

$<#Q.10G->#Q^#

R ³

^E

V = ( f 1 , f 2 )

$<#Q.10G-:#

^#

R ²

#%J@6uv0G??68(*2703Jc(8FI<6*(8<

]

03(*9>#

T : R ³ → R ²

^9:#7?9 ]

-:#7<BJ

]

#^10G<-@27#%-/.0G->#%- ?10G9

6Ž0+,03Jc9:(*2%#

M _U,V (T ) =

2 − 1 1 3 2 − 3

.

0Bw<j27FI<-:(*^)%9>#t6*#%-‚iI#%2%J>#7$9:-

e ⁰ 1 = e 2 + e 3

E

e ⁰ 2 = e 3 + e 1

#gJ

e ⁰ 3 = e 1 + e 2

k

‘}FI<BJ>9>#79@Da$#

U ⁰ = (e ⁰ 1 , e ⁰ 2 , e ⁰ 3 )

#7-bJ/#%<27FG9>#n$<#n.10G->#n^#

R ³

k‚" FG<<#79@6Ž0 +,03Jc9:(*27#

M _U 0 ,V (T )

^{9:#7?9 ]} -:#7<BJ50G<BJ

T

^10G<!-@68#7-;.103->#7-

U ⁰

^#gJ

V

^k

.xw<’27FG<->(8^)79:#“6*#%-=iG#72%J>#7$9:-

f 1 ⁰ = ¹ ₂ (f ¹ + f 2 )

^#gJ

f 2 ⁰ = ¹ ₂ (f ¹ − f 2 )

^k

Œ!0G2540G<BJ…Da$#

V ⁰ = (f 1 ⁰ , f 2 ⁰ )

#7-:J$<#.10G-:#GE3^FI<<#%96*0;+{03J>9>(827#

M _U 0 ,V ⁰ (T )

9>#%?9

]

->#%<aJc0G<BJ

T

^10G<-6*#%-.10G->#%-

U ⁰

^#%J

V ⁰

^k

”‡mˆ–•

?FI(*<BJc-b‹UŒ'FI(—J;6Ž0e+,03Jc9:(*2%#

J =









0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0







 .

0BwŒ'(1?FI->-:(*.68#GEa^(*0G˜IFI<1036*(*-:#79

J

k<[fFI$9><!(*9c0$!<#+{03J>9>(827#

S

#gJ6Ž0m+{0™Jc9>(827#

^(*0G˜IFI<1036*#

S ^{− 1} JS

27FG9>9>#%->?FI<^103<aJ>#Gk .xwš" FI<!<#79›68#œ?FI68AC<G+,#+,(8<(*+š$+’^#

J

#%Jž#%<e^

]

^!$(*9:#ŸDa$#

I, J, J ² , J ³ , J ⁴

->FG<aJ/6*(8<

]

03(*9>#%+,#%<BJŸ^

]

?S#%<^10G<BJc-%k

2WwŒ'FI(—J

P ∈ C [X]

$< ?FI6—A'<!I+,#h^#^!#7˜I9

]

0G$ ?68$-

3

k¡‘}FI<BJc9:#79eDa$#-:(

P (J) = 0

E1036*FI9:-

P = 0

^k

¢‡ ˆŠ£

?FI(*<BJ>-:‹!Œ'FI(—J$!<#‚+,03Jc9:(*2%#

A ∈ C ¹⁰ ₁₀

0WAB03<aJž?SFI$!9?FI68AC<G+,#2W039c0G2gJ

]

9:(*-bJc(*Da$#

#%J@+h(*<(8+$+¤9>#%->?#72%J>(8iG#7+h#7<BJ

χ _A (x) = (x − 1) ⁶ (x − 2) ⁴

^#%J

M _A (x) = (x − 1) ³ (x − 2) ² .

¥¦0§+,03J>9>(*2%#

A

#7-bJ:¨#76*68#ƒ^(Ž03˜IFI<10G68(*->0G.6*#,©ªo¬«G$-:J>(8€1#%95weŒ'(ŸJ>#76Ÿ<­uv#%-:J,?10G-e6*#ƒ270G-7E

^FI<<!#79JcFG$!Jc#%-6*#%- fFI9>+h#7- ^#h«GFI9>^0G<¡?SFG->->(8.6*#%- #gJ®5¯5¯N®g°1±¬²®g³´³*®gµ{®g°1±Ÿ¶G²*¯g±¬²´°­·g±|®5¯

¦‡mˆ–•

\]

M =









1 0 1 0 0 2 − 1 0 0 0 1 0 1 1 1 1







 .

<jfFG$9><(89c0e$<#t+,03Jc9:(*2%#

S

#gJ/6*0+{0™Jc9>(827#m9

]

^$(—Jc#

S ^{− 1} M S

27FI9:9>#7-:?SFG<^10G<BJc#3k

‡ ˆv£

?FI(*<BJc-b‹@Œ'FI(—J

E

^$< #7->?0G27#jiI#%2%JcFG9>(*#%6;^#j^(*+h#7<!->(*FG<“€1<(*#ƒ#gJ

T

^{$< #7<C¨}

^FI+hFI9>?!4(*-:+,#¡^#

E

^k ‘}FI<BJ>9>#79=#g&'?68(*2%(8Jc#%+,#%<BJjDC$!#

ker T ⊆ ker T ²

^#%J}DC$!#

Im T ⊇ Im T ²

^{k" ]} +hFI<BJc9:#79#7<!->$(—Jc#tDa$#

ker T = ker T ² ⇔ Im T = Im T ² .