+wˇow w 5nw n ,+w ˇ ˘w,55w , ˘ ww w w? u @4l ul? u *...

51
+WOWW 5NW N,+W w,55W , wWW W?_U@4L UL? U * U@4TL _i ?4ih hi@* UL? *i tL*|i LTih@3L? i UL? F * U@4TL _i ?4ih UL4T*itt N? ?4ihL UL4T*ittL i ?@ ULTT@ Lh_?@|@ _ ?4ih hi@* E@c K 5 U Uc UL i F iE@c KG @c K 5 Uj #i?@4L ; E@c K c ESc _ 5 F *i ti}i?| LTih@3L? G tL44@GE@c KnESc _’E@ n Kc S n _( ThL_L||LGE@c K ESc _’E@S K_c @_ n KS Vi?i ?@|h@*i _i?|U@hi @ 5 U UL? *@ ULTT@ E@c f 5 Fc UL i L}? ?4ihL hi@*i T L ittihi Ti?t@|L UL4i ?4ihL UL4T*ittL , F i ? R@4T* @4i?|L _ Uc UL i U F w<i*i4i?|L Efc 5 Fc t _i?L|@ UL? i_ i _i||L ?| @ 44@}?@h@ 5 @ ’ Efc Efc ’ Ec f ’ hi?_@4L E@c K 5 FG E@c K’E@c f n EfcK’E@c f n Efc EKc f ’ @ n Kc UL i L}? ?4ihL UL4T*ittL TL ittihi tUh||L ?i**@ uLh4@ @ n Kc _i||@ uLh4@ @*}iMhU@ @ i U@4@|@ T@h|i hi@*ic K i *@ T@h|i 44@}?@h@ wi LTih@3L? |h@ ?4ih UL4T*itt tUh|| ? uLh4@ @*}iMhU@ t iti}L?L UL? *i hi}L*i _i* U@*UL*L *i||ih@*iG E@ n KnES n _@ n K n S n _ ’E@ n Sn EK n _( E@ n K ES n _@S n @_ n KS n K_ @S n @_ n KS K_ ’E@S K_n E@_ n KS

Upload: others

Post on 13-Aug-2020

1 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

+W�OW��W 5NW �N�,+W ����w,55W , ��wW���W�

W?_�U��@4L UL? U �* U@4TL _i� ?�4ih� hi@*� UL? *i tL*�|i LTih@3�L?� i UL?F �* U@4TL _i� ?�4ih� UL4T*itt��

N? ?�4ihL UL4T*ittL i �?@ ULTT�@ Lh_�?@|@ _� ?�4ih� hi@*� E@c K� 5U� Uc U�Li

F ' iE@c K� G @c K 5 Uj�

#i�?�@4L ; E@c K� c ESc _� 5 F *i ti}�i?|� LTih@3�L?� G

�� tL44@G E@c K� n ESc _� ' E@n Kc Sn _�(��� ThL_L||LG E@c K� � ESc _� ' E@S� K_c @_n KS��

V�i?i ?@|�h@*i �_i?|��U@hi @ 5 U UL? *@ ULTT�@ E@c f� 5 Fc U�Li L}?�?�4ihL hi@*i T�L ittihi Ti?t@|L UL4i ?�4ihL UL4T*ittL , F i �? R@4T*��@4i?|L� _� Uc U�Li U � F�

w<i*i4i?|L Efc �� 5 Fc t� _i?L|@ UL? � i_ i _i||L �?�|@ �44@}�?@h�@�5� �@

�2 ' � � � ' Efc �� � Efc �� ' E��c f� ' ���

�hi?_�@4L E@c K� 5 FG

E@c K� ' E@c f� n Efc K� ' E@c f� n Efc �� � EKc f� ' @n �Kc

U�Li L}?� ?�4ihL UL4T*ittL T�L ittihi tUh�||L ?i**@ uLh4@ @ n �Kc _i||@uLh4@ @*}iMh�U@� @ i U��@4@|@ T@h|i hi@*ic �K i *@ T@h|i �44@}�?@h�@�wi LTih@3�L?� |h@ ?�4ih� UL4T*itt� tUh�||� �? uLh4@ @*}iMh�U@ t� iti}�L?L UL?*i hi}L*i _i* U@*UL*L *i||ih@*iG

E@n �K� n ESn �_� ' @n �Kn Sn �_ ' E@n S� n �EKn _�(E@n �K� � ESn �_� ' @Sn �@_n �KSn �2K_ ' @Sn �@_n �KS� K_

' E@S� K_� n �E@_n KS��

Page 2: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

#@|L 5 ' @n �Kc t� _i�?�tUi �* ?�4ihL UL4T*ittL UL?��}@|L

5 ' @� �K�

�ih ti4T*��U@hi �?@ uh@3�L?i UL? �? UL4T*ittL @ _i?L4�?@|Lhic t� T�L4L*|�T*�U@hi ?�4ih@|Lhi i _i?L4�?@|Lhi Tih �* UL?��}@|L _i* ?�4ihLc Lttih��@?_L U�i

E@n �K� � E@� �K� ' @2 � �@Kn �@K� �2K ' @2 n K2�

6�tt�@4L �? t�t|i4@ _� h�uih�4i?|L U@h|it�@?L �%+� N? ?�4ihL UL4T*ittLE@c K� T�L ittihi Ti?t@|L UL4i T�?|L _i* T�@?L Z ET�@?L UL4T*ittL�G @h@TThiti?|@ *<@tU�tt@ i K *<Lh_�?@|@�

�55,+V�~W��W��� 5i 5 5 U , 5 ' E@c f� , 5 t|@ t�**<@tti % E_i||L @tti hi@*i��2� 5i 5 ' �Kc U�Li 5 i �44@}�?@h�L T�hL , 5 ' Efc K� t|@ t�**<@tti + E_i||L

@tti �44@}�?@h�L��

6Lh4@ |h�}L?L4i|h�U@ _� �? ?�4ihL UL4T*ittL� 5�@ 5 ' @ n �K �? ?��4ihL UL4T*ittL i t�@ � *@ t�@ �44@}�?i t�* T�@?L Z� �ttLU�@4L @ 5 _�iRULLh_�?@|i TL*@h�� E4c w�G

�� 4 's@2 n K2 i _i||L 4L_�*L _� 5 i_ i �* ti}4i?|L U�i UL?}��?}i

*<Lh�}�?i UL? �* T�?|L � (2� w i _i||L @?L4@*�@ L @h}L4i?|L _� 5c i *<@?}L*L |h@ 4 i �* ti4�@tti

TLt�|��L _i**i % i_ i �?_���_�@|L @ 4i?L _� 4�*|�T*� _� 2Z�#@**@ |h�}L?L4i|h�@ t� �@ U�i

@ ' 4 ULt w K ' 4 t�? w

ULt w '@

4'

@s@2 n K2

t�? w 'K

4'

Ks@2 n K2

#@**i uLh4�*i ThiUi_i?|� t� L||�i?iG

2

Page 3: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

@n �K ' 4 EULt w n � t�? w� c

_i||@ uLh4@ |h�}L?L4i|h�U@ _i* ?�4ihL UL4T*ittL @n �K�#�i ?�4ih� UL4T*itt� ?L? ?�**� 5� ' 4�EULt w�n� t�? w�� i 52 ' 42EULt w2n

� t�? w2� tL?L �}�@*� ti 4� ' 42 i w� ' w2 n 2&Zc UL? & �?|ihL�

AiLhi4@� #@|� 5�c 52 5 Fc �* *LhL ThL_L||L i

5� � 52 ' 4�42 dULt Ew� n w2� n � t�? Ew� n w2�o�

A@*i hi}L*@ �@*i @?U�i Tih �? ?�4ihL ^�@*t�@t� _� u@||Lh�� 5i � u@||Lh� tL?L�}�@*�c t� �@ *@ uLh4�*@ _� #i �L��hiG

d4 EULt w n � t�? w�o? ' 4? EULt?w n � t�??w� c ? 5 Q�

5i t� UL?��i?i _� TLhhi

d4 EULt w n � t�? w�of ' �c Tih 4 9' f

d4 EULt w n � t�? w�o3? '�

d4 EULt w n � t�? w�o?c

*@ uLh4�*@ _� #i �L��hi �@*i Tih ? 5 ]�

+@_�U� ?�it�4i _i� ?�4ih� UL4T*itt�� 5�@ 5 5 F i ? 5 Q� W ?�4ih� � 5 F

|@*� U�i �? ' 5c t� _�UL?L h@_�U� ?�it�4i _� 5��hi?_�@4L 5 ' 4 EULt w n � t�? w� i t�@ � ' oEULt) n � t�?)� �?@ h@_�Ui

?�it�4@ _� 5� 5� _i�i @�ihi

�? ' 5 , o?EULt?)n � t�??)� ' 4 EULt w n � t�? w�, o? ' 4 i ?) ' w n 2&Zc & 5 ]

, o ' ?

s4 Eh@_�Ui @h�|4i|�U@� i ) '

w

?n

2&Z

?c & 5 ]�

w<�*|�4@ uLh4�*@ _@ �?�?�|� �@*Lh� Tih )c 4@ tL*L ? tL?L _�t|�?|��

AiLhi4@� 5 ' 4 EULt w n � t�? w� @44i||i tL*L ? h@_�U� ?�it�4i � _�t|�?|i_@|i _@**@ uLh4�*@

Page 4: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

Ey� � ' ?

s5W' ?

s4

�ULt

�w

?n

2&Z

?

�n � t�?

�w

?n

2&Z

?

��c

?i**@ ^�@*i t� TL?i & ' fc �c 2c ���c ?� ��Ew<@t|ih�tUL tih�i tL*L @ _�t|�?}�ihi *@ h@_�Ui UL4T*itt@ _@ ^�i**@ @h�|�

4i|�U@��B*� ? ?�4ih� uLh?�|� _@ Ey�c �@??L *L t|ittL 4L_�*L ?

s4c ^��?_� *i *LhL

�44@}�?� t�* T�@?L UL4T*ittL t|@??L t� �?@ U�hUL?uihi?3@ _� Ui?|hL *<Lh�}�?ii h@}}�L ?

s4�

5Ui}*�@4L 5 ' � ' �n f� ' �EULt fn � t�? f�� N|�*�33@?_L Ey�c |hL��@4L *i? h@_�U� ?�it�4i _i**<�?�|@G

"& '?

s�W

' ?

s�

�ULt

�f n

2&Z

?

�n � t�?

�f n

2&Z

?

��

' ULt2&Z

?n � t�?

2&Z

?c UL? & ' fc �c 2c ���c ?� ��

wi h@_�U� ?�it�4i _i**<�?�|@ t� h@TThiti?|@?L }iL4i|h�U@4i?|i UL4i �ih�|�U� _� �? TL*�}L?L hi}L*@hi _� ? *@|�c �?tUh�||L �? �?@ U�hUL?uihi?3@ _� h@}}�L�c UL? �? �ih|�Ui ?i* T�?|L E�c f��

,5,��W��� �@*UL*@hi *i h@_�U� U�M�U�i UL4T*itti _� H�5Uh���@4L H �? uLh4@ |h�}L?L4i|h�U@G

H ' H n f� ' HEULt f n � t�? f��

�& '�sHW

' �sH

�ULt

2&Z

�n � t�?

2&Z

�c & ' fc �c 2�

"��?_�

�f ' 2EULt f n � t�? f� ' 2(

�� ' 2

�ULt

2Z

�n � t�?

2Z

�' 2

#��

2n �

s�

2

$' �� n �

s�(

�2 ' 2

�ULt

eZ

�n � t�?

eZ

�' 2

#��

2� �

s�

2

$' ��� �

s��

e

Page 5: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

wi |hi h@_�U� U�M�U�i _� H tL?LG 2c �� n �s� i �� n �

s��

2� �@*UL*@hi *i h@_�U� ^�@h|i _� �� n �s��

5� �@G

4 's� n � ' 2c ULt w ' ��

2i t�? w '

s�

2, w '

2Z

, �� n �s� ' 2

�ULt

2Z

�n � t�?

2Z

��

�TT*�U@?_L *@ Ey� UL? & ' fc �c 2c �c t� �@

�& 'e

s�� n �

s�W

' es2

�ULt

�Z

Sn

&Z

2

�n � t�?

�Z

Sn

&Z

2

���

�f 'es2kULt

Z

Sn � t�?

Z

S

l' e

s2

#s�

2n �

2

$'

es2

2

�s� n �

�(

�� 'es2kULt

�ZSn

Z

2

�n � t�?

�ZSn

Z

2

�l' e

s2

�ULt

2Z

�n � t�?

2Z

' es2

#��

2n �

s�(

2

$'

es2

2

��� n �s��(

�2 'es2kULt

�ZSn Z

�n � t�?

�ZSn Z

�l' e

s2

�ULt

.Z

Sn � t�?

.Z

S

' es2

#�s�

2� �

2

$'

es2

2

��s�� �

�(

�� 'es2

�ULt

�Z

Sn

�Z

2

�n � t�?

�Z

Sn

�Z

2

��' e

s2

�ULt

DZ

�n � t�?

DZ

' es2

#�

2� �

s�(

2

$'

es2

2

��� �

s���

�� �@*UL*@hi *i h@_�U� tit|i _i**<�?�|@�5� �@G

"& 'Ss�W

' ULt2&Z

Sn � t�?

2&Z

Sc UL? & ' fc �c 2c �c ec D�

D

Page 6: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

"��?_�

"f ' ULt f n � t�? f ' �(

"� ' ULtZ

�n � t�?

Z

�'

2n �

s�

2(

"2 ' ULt2Z

�n � t�?

2Z

�' ��

2n �

s�

2(

"� ' ULtZ n � t�?Z ' ��(

"e ' ULteZ

�n � t�?

eZ

�' ��

2� �

s�

2(

"D ' ULtDZ

�n � t�?

DZ

�'

2� �

s�

2�

wi ti� h@_�U� |hL�@|i tL?L � �ih|�U� _� �? it@}L?L hi}L*@hi�

6Lh4�*@ _� ,�*ihL� �L?t�_ih�@4L *@ u�?3�L?i itTL?i?3�@*i _� �@h�@M�*iUL4T*itt@ s E5� ' e5c 5 5 F�

V@*}L?L *i UL?t�i|i ThLTh�i|@ _i**i TL|i?3iG

e5 � e� ' e5n�( e5 G e� ' e53�( Ee5�? ' e?5c

UL? 5c � 5 Fc ? 5 ]�5i 5 ' %n �+c UL? %c + 5 Uc t� TL?i

e5 ' e%n�+ ' e% � e�+ ' e% EULt + n � t�? +� �

"��?_� i

e�+ ' EULt + n � t�? +� �

"�it|<�*|�4@ i *@ uLh4�*@ _� ,�*ihL�5� �@

e2Z� ' ULt 2Z n � t�? 2Z ' � n f� ' �

i

S

Page 7: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

e5n2Z� ' e5 � e2Z� ' e5c

U�Li *<itTL?i?3�@*i e5 i Tih�L_�U@ _� Tih�L_L 2Z��W?L*|hi

e3�+ ' ULt E�+� n � t�? E�+� ' ULt + � � t�? +�

,5,��W��� ' f n �� ' ULt

Z

2n � t�?

Z

2' e

Z

2�(

�� ' �� n f� ' ULtZ n � t�?Z ' eZ�

N? TL*�?L4�L ?i**@ �@h�@M�*i % i �?<itThitt�L?i

� E%� ' @f n @�%n @2%2 n ���n @?%

?

UL? @fc @�c ���c @? 5 U EL F�c _i||� ULi�U�i?|� _i* TL*�?L4�L i ? 5 Qc_i||L }h@_L _i* TL*�?L4�LG ? ' _i}� E%�� 5i @? ' �c �* TL*�?L4�L i _i||L4L?�UL� w<�?t�i4i _i� TL*�?L4� �? % t� U EL F� i _i?L|@|L UL? U d%o ELF d%o�� W* TL*�?L4�L ?�**L i �* TL*�?L4�L UL? |�||� � ULi�U�i?|� �}�@*� @ f�N? TL*�?L4�L ULt|@?|i i �* TL*�?L4�L ?�**L LTT�hi �? TL*�?L4�L _� }h@_Lf�

#�i TL*�?L4� �? % tL?L �}�@*� ti �@??L �}�@*� � ULi�U�i?|� _i**i TL|i?3i_� % ULhh�tTL?_i?|�� W TL*�?L4� t� tL44@?L i t� 4L*|�T*�U@?L tiUL?_L *i hi}L*i?L|i�

�*}Lh�|4L _i**@ _���t�L?i� #@|� _�i TL*�?L4� �E%�c �E%� 5 U d%oc tL?L_i|ih4�?@|� �? 4L_L �?�UL � TL*�?L4� 'E%� E^�L3�i?|i� i -E%� Ehit|L� �?U d%oc |@*� U�i

�E%� ' �E%�'E%� n-E%�

UL? -E%� ' f LTT�hi _i}-E%� _i}�E%��5i _i}�E%� ' ?c _i}�E%� ' 6G�� ti 6 9 ? , _i}'E%� ' ?�6 i _i}-E%� 6(��� ti 6 : ? , _i}'E%� ' f i -E%� ' �E%��

.

Page 8: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

5i ?i* TL*�?L4�L �E%� 5 Ud%oc % i tLt|�|��|@ _@ �? S 5 Uc �* h�t�*|@|L i �?i*i4i?|L _� U Ei *L t|ittL �@*i ti �?�iUi _� U *@�Lh�@4L �? F��

,5,��W���E%� ' %� � 2%2 n 2 5 Ud%o�E�� ' 2.� �H n 2 ' �� 5 U�

AiLhi4@ _i* hit|L� 5�@?L �E%� 5 Ud%o i �E%� ' % � Sc UL? S 5 U� W*hit|L _i**@ _���t�L?i _� �E%� Tih �E%� i �ES��

,5,��W���E%� ' %� � 2%2 n 2c �E%� ' %� ��-E%� ' �E�� ' ���W* h�t�*|@|L t� T�L u@U�*4i?|i �ih��U@hi iti}�i?_L *@ _���t�L?i |h@ � _�i

TL*�?L4��

#@|� �E%�c �E%� 5 Ud%o UL? �E%� 9' fc _�U�@4L U�i �E%� i _���t�M�*i Tih�E%� ti �E%� ' �E%�'E%�c U�Li ti �* hit|L _i**@ _���t�L?i i f�

S 5 U t� _�Ui h@_�Ui EL 3ihL� _� �? TL*�?L4�L �E%� 5 Ud%o ti �ES� ' f�

AiLhi4@ _� +��?�� 5i �E%� 5 Ud%o i S 5 Uc @**Lh@ S i h@_�Ui _��E%� / �E%� i _���t�M�*i Tih %� S�

AiLhi4@� �}?� TL*�?L4�L � E%� 5 Fd%o _� }h@_L ?c ? A �c �@ it@||@4i?|i? h@_�U� �? F�

5i �* TL*�?L4�L i 4L?�UL i *i h@_�U� tL?L %�c ���c %? @**Lh@ TLtt�@4L tUL4�TLhhi �* TL*�?L4�L ?i**@ uLh4@

� E%� ' E%� %��E%� %2����E%� %?��

5i �?@ h@_�Ui UL4T@hi & �L*|i ?i**@ tUL4TLt�3�L?ic _�U�@4L U�i �@4L*|iT*�U�|@&�

�L? U2 �?_�U��@4L *<�?t�i4i _i**i ULTT�i E@c K� _� ?�4ih� hi@*�� 5� T�L_i�?�hi *@ tL44@ En� |h@ ULTT�iG

E@c K� n ESc _� ' E@n Sc Kn _�

i �* ThL_L||L E�� _� �? ?�4ihL hi@*i k Tih �?@ ULTT�@ E@c K�G

H

Page 9: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

k E@c K� ' Ek@c kK��

�L? U? EL F?� �?_�U��@4L *<�?t�i4i _i**i ?��T*i E@�c @2c ���c @?� _� ?�4ih�hi@*� EL UL4T*itt��� W? @?@*L}�@ UL? �* U@tL _� U2c t� _i�?�tUL?L �?@ tL44@i �? ThL_L||L Tih �? ?�4ihL hi@*i EL UL4T*ittL��

wi ThLTh�i|@ _� ^�it|i LTih@3�L?� tih�L?L UL4i |h@UU�@ Tih _i�?�hic T��@t|h@||@4i?|ic *@ ?L3�L?i _� tT@3�L �i||Lh�@*i�

5��~W V,AA�+W�wW

N?L tT@3�L �i||Lh�@*i i �? �?t�i4i T _� i*i4i?|� E�$� �c _i||� �i||Lh�ct� U�� tL?L _i�?�|i *i ti}�i?|� LTih@3�L?�G

�� �?@ tL44@G Tih �$� c�$� 5 T , �$� n�$� 5 T c |@*i U�iGE@� E�$� n�$� � n�$� ' �$� n E�$� n�$� �(EM� ,t�t|i

�$f 5 T |@*i U�iG

�$f n�$� ' �$� n

�$f ' �$� (

EU� ,t�t|i ��$� 5 T |@*i U�iG

�$� n E��$� � ' E��$� � n�$� ' f(

E_� �$� n�$� ' �$� n�$� c

Tih L}?� �$� c�$� c�$� 5 T �2� �? ThL_L||L b�$� _� �? �i||Lhi�$� Tih �? ?�4ihL b hi@*i EL UL4T*ittL�c

|@*i U�iGE@� b E�$� n�$� � ' b�$� n b�$� (EM� Ebn >��$� ' b�$� n >�$� (EU� Eb>��$� ' b E>�$� �(E_� ��$� ' �$� cTih L}?� bc > 5 Uc�$� c�$� 5 T �

A�||i ^�it|i ThLTh�i|@ tL?L �ih��U@|i Tih }*� �?t�i4� U? L F? UL? *iLTih@3�L?� U�|@|i ?i* T@h@}h@uL ThiUi_i?|ic �? T@h|�UL*@hic Efc fc ���c f� u�?}i_@

�$f i E�%�c�%2c ���c�%?� u�?}i _@ � E%�c %2c ���c %?���*U�?i ThLTh�i|@ �?|ihitt@?|� U�i t� h�U@�@?L _@**i _i�?�3�L?� tL?L *i

ti}�i?|�G

b

Page 10: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

f�$� '�$f i ��$� ' E����$� �

W?u@||�c Tih L}?� �$� 5 T c

f�$� n�$� ' Ef n ���$� ' ��$� ' �$�

i ^��?_�

f�$� '�$f c

TihU�i *<i*i4i?|L�$f i �?�UL� EW?u@||�c Tih _i�?�3�L?i _�

�$f c t� �@ f�$� n�$

f ' f�$� i _@**@ hi*@3�L?i f�$� n �$� ' �$� @TTi?@ _�4Lt|h@|@c tL44@?_L��$� c t� �@

Ef�$� n�$� � n E��$� � ' �$� n E��$� � '�$f

o

f�$� n E�$� n E��$� �� ' f�$� n�$f �

#@ f�$� n�$f ' f�$� i f�$� n

�$f '

�$f c ti}�i f�$� '

�$f ��

W?L*|hic

E�b��$� n b�$� ' E�bn b��$� ' f�$� '�$f c

U�Li E�b��$� ' �b�$� Ei *<�?�ihtL _� b�$� ��

,5,��W���L?t�_ih�@4L U EL F� i t�@ U d%o EL F d%o� *<�?t�i4i _i� TL*�?L4� �? % @

ULi�U�i?|� �? UG

U d%o ' i� E%� ' @f n @�%n ���n @?%? G @fc @�c ���c @? 5 Uj �

�f

Page 11: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

U d%o i �?L tT@3�L �i||Lh�@*i� W?u@||�c _@|�

sE%� ' @f n @�%n ���n @?%?c }E%� ' Kf n K�%n ���n K6%

6 5 U d%o

i b 5 Uc �* TL*�?L4�L tL44@

sE%�n }E%� ' E@fn Kf�nE@�n K��%n ���nE@?n K?�%?n K?n�%

?n�n ���n K6%6

i �* TL*�?L4�L ThL_L||L

bsE%� ' b@f n b@�%n ���n b@?%?

@TT@h|i?}L?L @ U d%o�

5�AA�5��~W c #W�,�#,�~� wW�,�+, , ��5W

N? tL||L�?t�i4i ` _� T i _i||L tL||LtT@3�L �i||Lh�@*i _� T ti i ittLt|ittL �?L tT@3�L �i||Lh�@*i EUL? *i t|itti LTih@3�L?� _� T �� W? @*|hi T@hL*ic` i �? tL||LtT@3�L �i||Lh�@*i ti

� ��$f 5 ` (

2� ti �$� 5 ` @?U�i ��$� 5 ` (�� ti �$� c�$� 5 ` @?U�i �$� n�$� 5 ` (e� ti b 5 U EL F� i �$� 5 ` @?U�i b�$� 5 ` �

,5,��W���L?t�_ih�@4L � tL||L�?t�i4� _� U2

` ' iE@c @� UL? @ 5 Uj~ ' iE@c �� UL? @ 5 Uj�

` i �? tL||LtT@3�L _� U2c ~ ?L? i �? tL||LtT@3�Lc TihU�i

E@c �� n EKc �� ' E@n Kc 2� *5 ~�

��

Page 12: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

� T@h|�hi _@ L i ` tL||LtT@3� _� T c ULt|h��@4L @*|h� tL||LtT@3� _� T G@� w<�?|ihti3�L?i _� L i ` i ULt�|��|@ _@� �i||Lh� U�i t|@??L t�@ �? L U�i

�? `

L _` ' i�$� 5 T G �$� 5 L i �$� 5 `j �

L _` i �? tL||LtT@3�L _� T c 4i?|hi L ^` �? }i?ihi ?L? *L i��_ iti4T�Lc t�@?L L i` _�i hi||i �? U2� w<�?�L?i _� L i` ?L? UL?|�i?i

*@ tL44@ _� �? i*i4i?|L _� L i �?L _� ` _��iht� _@�$f �

�ih u@MMh�U@hi �? tL||LtT@3�Lc t� @}}��?}L?L |�||� }*� i*i4i?|� 4@?U@?|��M� �L?t�_ih�@4L *L tT@3�L uLh4@|L _@ |�||i *i tL44i _� i*i4i?|� _� L

UL? i*i4i?|� _� ` G

L n` ' i�$� n�$� UL? �$� 5 L c �$� 5 `j�

L n` i �? tL||LtT@3�L �i||Lh�@*i _� T �

5i L _ ` 'q�$frc *< LTih@3�L?i n |h@ tL||LtT@3� t� �?_�U@ UL? L � `

i t� _�Ui tL44@ _�hi||@( *@ ULt@ �?|ihitt@?|i i U�ic �? ^�it|L U@tLc L}?�i*i4i?|L _� L �` t� tUh��i �? 4L_L �?�UL UL4i tL44@ _� �? i*i4i?|L _�L i �?L _� ` �

,5,��W� E8���L?t�_ih�@4L �? U

� � ti}�i?|� tL||LtT@3�G

L ' iE%c fc %� G % 5 Uj(T ' iE%c fc f� G % 5 Uj(

` ' iE%c fc +� G %c + 5 Uj(~ ' iEfc %c +� G %c + 5 Uj�

5� �@

L _ T ' iE%c fc %� ' E%c fc f�j ' iEfc fc f�j(L _` ' iE%c fc %� ' E%c fc +�j ' iE%c fc %�j ' L (

~ _` ' iEfc %c +� ' E%c fc +�j ' iEfc fc +�j(L n T ' iE%c fc %� n E%�c fc f� UL? %c %�5 Uj

' iE@c fc K� UL? @c K 5 Uj ' ` �

�2

Page 13: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

5�@?L �$� �c ���c�$� ? �i||Lh� _� �?L tT@3�L �i||Lh�@*i T i @�c ���c @? 5 U EL F��W* �i||Lhi

�$� ' @��$� � n ���n @?�$� ? '?S�'�

@��$� �c

t� _�Ui UL4M�?@3�L?i *�?i@hi _i� �i||Lh� �$� �c ���c�$� ?�5�@ 7 ' i�$� �c ���c�$� ?j �? �?t�i4i �?�|L _� �i||Lh� _� T � wL tT@3�L

5T@?i�$� �c ���c�$� ?j ' i|�||i *i TLtt�M�*� UL4M�?@3�L?� *�?i@h� _� �$� �c ���c�$� ?j

i _i||L tT@? *�?i@hi i_ i �? tL||LtT@3�L �i||Lh�@*i _� T c ThiU�t@4i?|i i�* T�� T�UUL*L tL||LtT@3�L U�i UL?|�i?i � �i||Lh� �$� �c ���c�$� ?�

#�hi4L U�i�$� �c ���c�$� ? }i?ih@?L *L tT@3�L �i||Lh�@*i T L U�i i�$� �c ���c�$� ?ji �? t�t|i4@ _� }i?ih@|Lh� _� T ti T ' 5T@?i�$� �c ���c�$� ?j� "��?_�i�$� �c ���c�$� ?j i �? t�t|i4@ _� }i?ih@|Lh� _� T ti i tL*L ti ;�$� 5 T it�t|L?L@�c ���c @? 5 U EL F� |@*� U�i �$� ' @��$� � n ���n @?�$� ?�

W �i||Lh��$� �c ���c�$� ? 5 T t� _�UL?L *�?i@h4i?|i _�Ti?_i?|� ti <@�c ���c @? 5U EL F� ?L? |�||� ?�**�c |@*� U�i @��$� � n ��� n @?�$� ? '

�$f � �*|h�4i?|�c t�

_�UL?L *�?i@h4i?|i �?_�Ti?_i?|� EU�Li @��$� � n ��� n @?�$� ? '�$f , @� '

fc ;���N? �?t�i4i �?�|L i�$� �c ���c�$� ?j _� �i||Lh� _� T c t� _�Ui M@ti _� T ti�$� �c ���c�$� ? tL?L *�?i@h4i?|i �?_�Ti?_i?|� i ti tL?L �? t�t|i4@ _�

}i?ih@|Lh� _� T � 5i i�$� �c ���c�$� ?j i �?@ M@ti _� T c @**Lh@�� �$� �c ���c�$� ? }i?ih@?L T , ;�$� 5 Tc�$� ' @��$� � n ���n @?�$� ?(2� @�c ���c @? tL?L _i|ih4�?@|� �? 4L_L �?�UL _@ �$� EU�L i _L��|L @* u@||L

U�i �$� �c ���c�$� ? tL?L *�?ih4i?|i �?_�Ti?_i?|���@�c ���c @? 5 U ELF� t� _�UL?L ULLh_�?@|i _��$� h�tTi||L @**@ M@ti�$� �c ���c�$� ?�5i �?L tT@3�L �i||Lh�@*i T TLtt�i_i �?@ M@ti �?�|@ i�$� �c ���c�$� ?jc �* ?��

4ihL ? t� _�Ui _�4i?t�L?i _� T i t� �?_�U@ UL? _�4T � 5i T 'q�$frc t�

TL?i _�4T ' f� 5i T 'q�$frc LTT�hi T TLtt�i_i �?@ M@ti �?�|@c _�U�@4L

U�i T �@ _�4i?t�L?i �?�|@�

,5,��W��� W? U2c UL?t�_ih�@4L � �i||Lh�

�$e � ' E�c f�c�$e 2 ' Efc ���

��

Page 14: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

�$e �c�$e 2 }i?ih@?L U2� W?u@||�c t�@ �$� ' E��c �2� 5 U2� 5� �@

�$� ' E��c �2� ' E��c f� n Efc �2� ' ��E�c f� n �2Efc �� ' ���$e � n �2�$e 2�

W?L*|hic E��c �2� '�$f / �� ' �2 ' f , �$e �c�$e 2 *�?i@h4i?|i �?_�Ti?_i?|�

, i�$e �c�$e 2j i �?@ M@ti _� U2 i U2 �@ _�4i?t�L?i 2� i�$e �c�$e 2j t� U��@4@M@ti U@?L?�U@ _� U2�

2� W? U�c *@ M@ti U@?L?�U@ i ULt|�|��|@ _@� |hi �i||Lh�

�$e � ' E�c fc f�c�$e 2 ' Efc �c f�c�$e � ' Efc fc ���

�� W* tL||LtT@3�L U d%o9�

_i� TL*�?L4� _� }h@_L 9 �c TLtt�i_i �?@ M@ti�?�|@G i�c %c %2c %�j�

W?u@||�c �* TL*�?L4�L

� E%� ' @f n @�%n @2%2 n @�%

� ' @f � � n @� � %n @2 � %2 n @� � %�

i UL4M�?@3�L?i *�?i@hi _� i�c %c %2c %�j , i�c %c %2c %�j i �? t�t|i4@ _�}i?ih@|Lh�� W?L*|hic ti

@f n @�%n @2%2 n @�%

� ' f , @f ' @� ' @2 ' @� ' fc

^��?_� �c %c %2c %� tL?L *�?i@h4i?|i �?_�Ti?_i?|�� U d%o9�

�@ _�4i?t�L?ie E?�4ihL _� i*i4i?|� _i**@ M@ti��

5i L �@ M@ti �$� �c ���c�$� , i T �@ M@ti �$e �c ���c�$e 6c @**Lh@

L n T ' 5T@?i�$� �c ���c�$� ,c�$e �c ���c�$e 6j�

AiLhi4@ _i**i _�4i?t�L?�� 5i L i T tL?L tT@3� �i||Lh�@*� _� _�4i?�t�L?i �?�|@c @**Lh@

_�4EL n T � ' _�4L n _�4T � _�4EL _ T ��

�e

Page 15: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

W? T@h|�UL*@hi Eh�ULh_@?_L U�i Tih *@ tL44@ _�hi||@ L �T t� �@ L _T 'q�$fr�c

_�4EL � T � ' _�4L n _�4T �

,5,��W��Vi_�@4L U�i Tih }*� tT@3� _i**<iti4T�L E8� i tL__�tu@||L �* |iLhi4@ _i**i

_�4i?t�L?�� 5� �@G L _ T ' iEfc fc f�j , _�4EL _ T � ' f�L i uLh4@|L _@ �i||Lh� _i* |�TL E%c fc %� ' %E�c fc ��c U�Li _@ 4�*|�T*� _�

E�c fc ��c ^��?_� i }i?ih@|L _@ E�c fc �� , _�4L ' ��T i uLh4@|L _@ �i||Lh� _i* |�TL E%c fc f� ' %E�c fc f�c ^��?_� i }i?ih@|L _@*

�i||Lhi E�c fc f� , _�4T ' ��N? �i||Lhi _� ` ' L n T ic �?�iUic _@|L _@ @E�c fc f� n KEfc fc ��c U�Li i

UL4M�?@3�L?i *�?i@hi _i� �i||Lh� E�c fc f� i Efc fc �� , _�4` ' 2� �**Lh@ �@*i_�4` ' _�4L n _�4T � _�4EL _ T ��

�+�#�AA� 5��w�+, +,�w, N5N�w,

�L?t�_ih�@4L U? UL? *@ M@ti U@?L?�U@ �$e �c ���c�$e ? E�$e � ' E�c fc ���c f� c�$e 2 ' Efc �c fc ���c f� c ���c�$e ? ' Efc ���c fc ��� i t�@?L �$� c�$� 5 U? _�i �i||Lh�G�$� ' E��c �2c ���c �?�c �$� ' E��c �2c ���c �?�� #i�?�@4L �* ThL_L||L tU@*@hi _��$� i �$� G

�$� � �$� '?S�'�

�����

�hLTh�i|@G ;�$� c�$� c�$� 5 U?c @c K 5 UG�� E@�$� n K�$� � � �$� ' @�$� � �$� n K�$� � �$� (2� �$� � E@�$� n K�$� � ' @�$� � �$� n K�$� � �$� (�� �$� � �$� A f i �$� � �$� ' f / �$� '

�$f (s�$� � �$� t� �?_�U@ UL?n�$� n i t� _�Ui ?Lh4@ L *�?}�i33@ _� �$� �

e��$� ��$� ' n�$� n n�$� n ULt wc _L�i w i *<@?}L*L UL4ThitL |h@ � _�i�i||Lh� �$� i �$� �

#�i �i||Lh� �$� i �$� t� _�UL?L Lh|L}L?@*� ti �$� � �$� ' f�5i T i �?L tT@3�L �i||Lh�@*i i ` �? t�L tL||LtT@3�L ?L? ��L|Lc *<�?t�i4i

_i� �i||Lh� Lh|L}L?@*� @_ L}?� �$� 5 ` t� _i?L|@ UL? `zG

�D

Page 16: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

`z ' i�$� 5 T G �$� � �$� ' fc;�$� 5 `j�

`z i �? tL||LtT@3�L �i||Lh�@*i _� T U��@4@|L �* tL||LtT@3�L Lh|L}L?@*i @

` � W?L*|hic ` _`z 'q�$fri ` n`z ' T c U�Li T ' ` �`z� "��?_�c

Tih �* |iLhi4@ _i**i _�4i?t�L?�c _�4T ' _�4` n _�4`z�

,5,��W���L?t�_ih�@4L �? U� � �i||Lh�

�$� � '

3C �

��

4D c�$� 2 '

3C f

���

4D c�$� � '

3C �

2f

4D c�$� e '

3C �

f�2

4D c

�$� D '

3C f

22

4D c�$� S '

3C 2

22

4Dc

i � tL||LtT@3� L ' 5T@?i�$� �c�$� 2j i T ' 5T@?i�$� �c�$� ec�$� Dc�$� Sj��@*UL*�@4L *@ _�4i?t�L?i i �?@ M@ti Tih � tL||LtT@3� L _ T i L n T �5� ?L|@ t�M�|L U�i �$� �c�$� 2c�$� e tL?L �?_�Ti?_i?|� i U�i �$� �c�$� Dc�$� S tL?L

*LhL UL4M�?@3�L?� *�?i@h�G �$� S ' 2�$� �c �$� � ' �$� �n�$� 2 i �$� D ' �$� �n�$� 2��$� e� "��?_�c T ' 5T@?i�$� �c�$� 2c�$� ej� �i ti}�i U�i L_T ' 5T@?i�$� �c�$� 2j�@ _�4i?t�L?i _�i i �?@ M@ti i �$� �c�$� 2� L n T ' 5T@?i�$� �c�$� 2c�$� ej �@_�4i?t�L?i |hi i �?@ M@ti i �$� �c�$� 2c�$� e�

�ih tUh��ihi �?@ M@ti _i* tL||LtT@3�L Lh|L}L?@*i @ L _ T c U�i �@ _�4i?�t�L?i �c M@t|@ tUh��ihi �? �i||Lhi E@c Kc S� Lh|L}L?@*i @ �$� �c�$� 2� #i�i ittihiG

�@n Kn S ' fK� S ' f

,;?=

@ ' �2SK ' SS ' |

, E@c Kc S� ' |E�2c �c ���

"��?_�c Tih | ' �c �?@ M@ti _� EL _ T �z i _@|@ _@* �i||Lhi E�2c �c ���

+,AA, , �W��W W� U?

�S

Page 17: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

5�@ �$� 5 U? i t�@ �$% f �? T�?|L _� U?� 5� _�Ui hi||@ _� _�hi3�L?i �$�T@tt@?|i Tih �$% f �* tL||L�?t�i4i

o ' i�$% f n |�$� UL? �4 | 4j�

W? U2c �$� ' E�%c �+�c �$% f ' E%fc +f� i o ' E%c +�� wi i^�@3�L?� T@h@�4i|h�U�i _� o tL?LG

�% ' %f n |�%+ ' +f n |�+

�?@*L}@4i?|ic *i i^�@3�L?� T@h@4i|h�U�i _� �?@ hi||@ �? U� tL?LG

;?=

% ' %f n |�%+ ' +f n |�+5 ' 5f n |�5

5� _�Ui T�@?L �? U� _� _�hi3�L?� �$� i �$� T@tt@?|i Tih �$% f �* tL||L�?�t�i4i

� ' i�$% f n |�$� n r�$� UL? �4 |c r 4jc

UL? �$% fc�$� c �$� 5 U� i �$� c�$� �?_�Ti?_i?|�� wi i^�@3�L?� T@h@4i|h�U�i_� �? T�@?L tL?L _@|i _@G

;?=

% ' %f n |�% n r�%

+ ' +f n |�+ n r�+

5 ' 5f n |�5 n r�5

5� _�Ui 6�T�@?L _� U? _� _�hi3�L?� �?_�Ti?_i?|� �$� �c ���c�$� 6 E6 ?�T@tt@?|i Tih �$% f �* tL||L�?t�i4i

� '

��$% f n

6S�'�

|��$� � UL? �4 | 4��

�.

Page 18: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

#�U�@4L U�i *i hi||i o ' �$R n |�$� i o�' �$̂n |�$��tL?LGT@h@**i*i ti �$� ' �$��(Lh|L}L?@*� ti �$� � �$��' f(�?U�_i?|� ti o _ o� 9' >(UL4T*@?@h� ti <Z |@*i U�i o � Z i o�� Z�

,5,��W���L?t�_ih�@4L *i hi||i

o ' Efc fc �� n|s2E�c��c f�

o���' E�c fc f� n|s2E�c��c f�

o�' Efc �c f� n | Efc fc ���

�L�U�i � ' ����c o i o���tL?L T@h@**i*i� W?L*|hi ��� � ' Efc fc �� � E�c��c f� ' fc^��?_� o�i Lh|L}L?@*i @ o�

Vi_�@4L ti o i o�tL?L �?U�_i?|�� �� U��i_�@4LG ^�@?_L �? }i?ih�UL � 5 oi �}�@*i @ ��5 o�q �L?�@4LG

Efc fc �� n|s2E�c��c f� ' Efc �c f� n | Efc fc ��

,�

|s2c� |s

2c �

�' Efc �c |�

, |s2' f i | ' � , �4TLtt�M�*i-

wi hi||i o i o�?L? tL?L �?U�_i?|�� AhL��@4L �?@ hi||@ o��T@h@**i*@ @ o�i�?U�_i?|i UL? o� �L�U�i _i�i ittihi o��T@h@**i*@ @ o�c t� @�h@

o��' E@c Kc S� n |Efc fc ���

�ih |hL�@hi E@c Kc S�c �4TL?�@4L U�i o t�@ �?U�_i?|i UL? o�G

�|s2c� |s

2c �

�' E@c Kc Sn |��

�H

Page 19: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

�hi?_�@4L | ' f� �MM�@4L

Efc fc �� ' E@c Kc S� , o��' Efc fc �� n | Efc fc ���

VL}*�@4L UihU@hi Lh@ �? T�@?L U�i UL?|i?}@ o i o���� 5�@

Z ' E@c Kc S� n |�� n r�2�

Z _i�i UL?|i?ihi o , Z ' Efc fc �� n|s2E�c��c f� n r E@c Kc S��

�ih |hL�@hi E@c Kc S�c �4TL?�@4L U�i UL?|i?}@ o����

E�c fc f� n|s2E�c��c f� ' Efc fc �� n

|s2E�c��c f� n r E@c Kc S�

, E�c fc f� ' Efc fc �� n r E@c Kc S�

6�tt�@4L �? r TLtt�M�*ic Tih iti4T�L r ' �G

E�c fc f� ' E@c Kc Sn �� , @ ' �c K ' fc S ' ��

, Z ' Efc fc �� n|s2E�c��c f� n

rs2E�c fc����

�55,+V�~W��W�#�i hi||i �?U�_i?|� tL?L UL4T*@?@h��#�i hi||i ?L? �?U�_i?|� tL?L UL4T*@?@h� tL*L ti T@h@**i*i�

,5,��W��wi hi||i o i_ o��_i**<iti4T�L ThiUi_i?|i tL?L �?U�_i?|�c ^��?_� UL4T*@?@h��

AhL��@4L �? T�@?L U�i *i UL?|�i?i�

o ' Efc fc �� n|s2E�c��c f�

o��' Efc fc �� n | Efc fc ���

N? T�@?L U�i *i UL?|�i?i ic @_ iti4T�Lc �* T�@?L }i?ih@|L _@� �i||Lh�_�hi3�L?i _i**i _�i hi||iG

�b

Page 20: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

Z ' Efc fc �� n|s2E�c��c f� n rEfc fc ���

#�i T�@?� Z i Z�tL?L T@h@**i*� ti Z _ Z�' >� 5i

Z ' Rn |�� n r�2 i Z�' R�n |��� n r��2c

@**Lh@ Z i Z�tL?L T@h@**i*� ti

5T@?i��c �2j ' 5T@?i���c ��2j i R� R� *55T@?i��c �2jc

U�Li ti Z�' Z n ^c ^ 9' Efc fc ���c f��

,5,��W��L?t�_ih�@4L � T�@?�

Z ' Efc fc f� n |E�c fc f� nrs2E�c fc ��

Z�' Efc �c f� n |E�c fc f� n rEfc fc ���

5� �@

5T@?i��c �2j ' i� ' @�� n K�2 ' @E�c fc f� n KE�c fc �� ' E@n Kc fc K�j5T@?i���c ��2j ' i� ' @���� n K���2 ' @�E�c fc f� n K�Efc fc �� ' E@�c fc K��j

W _�i tT@3� UL�?U�_L?LG M@t|@ Thi?_ihi K�' Kc @�' @nKc ^��?_� � _�i T�@?�tL?L T@h@**i*��

W?�iUic

Z ' Efc fc f� n |E�c fc f� nrs2E�c fc ��

Z��' Efc fc �� n |��E�c fc f� n r��Efc �c f�

?L? tL?L T@h@**i*� i Z _ Z��' o� W?u@||�c Z ' Z��ti

2f

Page 21: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

�|n

rs2c fc

rs2

�' E|��c r��c ��c

U�Li ti r��' fc r 's2c |��' |n�� �**Lh@ � T�?|� _i* |�TL E|n �c fc �� 5 Z_Z��c

U�Li *@ hi||@

o ' E�c fc �� n |E�c fc f�

i *<�?|ihti3�L?i _i� _�i T�@?��

#�i T�@?� Z ' R n |�� n r�2 i Z�' ^ n |��� n r��2 tL?L Lh|L}L?@*� ti *i*LhL _�hi3�L?� ?Lh4@*� tL?L Lh|L}L?@*��

,5,��W��#@|L

Z ' Efc fc f� n |E�c fc f� n rE�c fc ��c

UL? �� ' E�c fc f� i �2 ' E�c fc �� E�2 ?L? ?Lh4@*�33@|L�c U@*UL*�@4L *@_�hi3�L?i �$? ?Lh4@*i @ Z� �Lt|L �$? ' E@c Kc S�c _i�i ittihi

E@c Kc S� � E�c fc f� ' @ ' fE@c Kc S� � E�c fc �� ' @n S ' f , S ' fc

^��?_� �$? ' |Efc �c f�� �L?t�_ih�@4Lc �?L*|hic �* T�@?L

Z�' Efc fc f� n |Efc �c f� n rE2c fc ���

�@*UL*�@4L *@ t�@ ?Lh4@*i�$?�G

E@�c K�c S�� � Efc �c f� ' K�' fE@�c K�c S�� � E2c fc �� ' 2@�n S�' f , S�' �2@�

, �$?� ' |�E��c fc 2��

2�

Page 22: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

�h@ �$? � �$?� ' Efc �c f� � E��c fc 2� ' f , �$? B �$?� , ZcZ�Lh|L}L?@*��

,5,��W��#@|L �* T�@?L

Z ' Efc fc f� n |E�c fc f� n rE�c fc ��c

|hL�@hi *@ hi||@ T@tt@?|i Tih Efc fc f� i Lh|L}L?@*i @* T�@?L Z�w@ hi||@ UihU@|@ �@ i^�@3�L?i

o ' Efc fc f� n |E@c Kc S�

i *@ _�hi3�L?i E@c Kc S� _i�i ittihi Lh|L}L?@*i @ E�c fc f� i @ E�c fc ��� 5� _i�i^��?_� @�ihi

E�c fc f� � E@c Kc S� ' @ ' fcE�c fc �� � E@c Kc S� ' @n S ' f , S ' f�

�**Lh@ o ' Efc fc f� n |Efc �c f��#i|ih4�?�@4L Z� o� �ttih��@4L U�i E�c fc f�c E�c fc �� i Efc �c f� tL?L |hi

�i||Lh� �?_�Ti?_i?|�c Tih U��

Z � o ' U��

��A+W�W

�L?t�_ih�@4L *<�?t�i4i

�E?�6� '

�|@Mi**i _� ?6 ?�4ih� Ehi@*� L UL4T*itt��

Lh_�?@|� �? ? h�}�i i �? 6 UL*L??i

22

Page 23: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

' i4@|h�U� @ ? h�}�i i_ 6 UL*L??ij

+@TThiti?|�@4L �?@ 4@|h�Ui UL4i �?@ |@Mi**@ _i**@ uLh4@G

3EEEC

@�� @�2 ��� @�?@2� @22 ��� @2?���

���� � �

���@?� @?2 ��� @??

4FFFD c

*@ �?_�U��@4L UL? � i tUh���@4L � ' E@���c _L�i @�� i �* }i?ih�UL i*i4i?|L_� � UL?|h@tti}?@|L _@ �? �?_�Ui _� h�}@ � i �? �?_�Ui _� UL*L??@ �� 5i 6 ' ?c*@ 4@|h�Ui t� _�Ui ^�@_h@|@ _� Lh_�?i ?� 5i �c� 5 �E?�6� i b 5 Uc t�TLttL?L _i�?�hiG

�� �n� ' � 5 �E?�6�c tL44@?_L }*� i*i4i?|� ?i**L t|ittL TLt|LS�� ' @�� n K��2� b� 5 �E?�6�c UL? Eb���� ' b@���

5i � 5 �E?�6� i � 5 �E6� ^�c t� _i�?�tUi �* ThL_L||L |h@ 4@|h�U�

�� ' � 5 �E?� ^�

UL? �* ThL_L||L h�}�i Tih UL*L??iG E����&c U�Li *<i*i4i?|L _� TLt|L �c &_i**@ 4@|h�Ui ��c i �* ThL_L||L _i**@ ��it�4@ h�}@ _� �c E@��@�2���@�?�c Tih *@

&�it�4@ UL*L??@ _� �c

3EC

K�&���

K6&

4FDG

E@��@�2���@�?�

3EC

K�&���

K6&

4FD ' @��K�& n @�2K2& n ���n @�6K6&�

W* }i?ih�UL i*i4i?|L _� � iG S�� 'S6

@�6K6�� �L|�@4L U�i � i � TLttL?L

ittihi 4L*|�T*�U@|i |h@ *LhL tL*L ti

2�

Page 24: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

?�4ihL UL*L??i _� � ' ?�4ihL h�}�i _� ��

#�i 4@|h�U� ^�@_h@|i _i**L t|ittL Lh_�?i t� TLttL?L ti4Thi 4L*|�T*�U@hi�

,5,��W��E2� 2�

�@ KS _

��e s} �

�'

�@en K} @s n K�Sen _} Ss n _�

��

�55,+V�~W��, E�4TLh|@?|i-��5i t� TLttL?L u@hi t�@ �� U�i ��c �? }i?ih@*i t� �@ U�i �� 9' ���

W?L*|hic �� ' �c _L�i � '

�f ff f

�c ?L? �4T*�U@ U�i � L � t�@?L 3ihLG

�� ff f

��f f� f

�'

�f ff f

��

w@ 4@|h�Ui |h@tTLt|@ �| _� � i *@ 4@|h�Ui L||i?�|@ _@ � tU@4M�@?_L *i

h�}�i UL? *i UL*L??i� 5i � '

�@ KS _

�c @**Lh@ �| '

�@ SK _

�� W? }i?ih@*i

E�|��& ' @&��

�hLTh�i|@ �44i_�@|iG�� E�n��| ' �| n�|(2� E���| ' �|�|�

5i � ' E@��� i �?@ 4@|h�Ui ^�@_h@|@c }*� i*i4i?|� @��c @22c ���c @?? ULt|�|���tUL?L *@ _�@}L?@*i Th�?U�T@*i _� �� N?@ 4@|h�Ui t� _�Ui _�@}L?@*i ti }*�i*i4i?|� u�Lh� _i**@ _�@}L?@*i Th�?U�T@*i tL?L ?�**��

�L?t�_ih�@4L 4@|h�U� ^�@_h@|i ETih Lh@ 2 � 2�� w@ 4@|h�Ui UL? |�||� �

t�**@ _�@}L?@*i Th�?U�T@*i i *@ 4@|h�Ui �_i?|�|@ U '

�� ff �

�i_ �@ *i

t|itti ThLTh�i|@ _i* ?�4ihL �c U�Li �U ' U� ' ��5@TT�@4L U�i ti @ 5 U i @ 9' fc TLtt�@4L _i�?�hi �? �?�ihtL @3� |@*i U�i

@@3� ' �� �i* U@tL _i**i 4@|h�U�c ?L? M@t|@ U�i � 9' � Tih TL|ih _i�?�hi �?@�3� |@*i U�i ��3� ' �3�� ' U� 5ih�i *@ ?L3�L?i _� _i|ih4�?@?|i _i**@4@|h�Ui � E_i|��� �i* U@tL 2� 2c t� �@

2e

Page 25: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

_i|

�@ KS _

�' @_� KS�

5� �i_i t�M�|L U�i _i| E��� ' _i|�_i|�� 5i _i|� 9' fc t� T�L _i�?�hi�3�G

ti � '

�@ KS _

�c t� �@ �3� '

_i|�

�_ �K�S @

��

5� �ih��U@ �44i_�@|@4i?|i U�i ��3� ' �3�� ' U�

�hLTh�i|@ _i**<�?�iht@G @� _i|�3� '�

_i|�(

M� E�3��3�

' �(U� E���3� ' �3��3�(_� E�|�3� ' E�3��|�

5� ?L|� @?U�i U�i _i|�| ' _i|���ih *i 4@|h�U� ?�?c �* _i|ih4�?@?|i t� _i�?�tUi h�ULht��@4i?|i ?i* ti}�i?|i

4L_LG ti

� '

3EEEC

@�� @�2 ��� @�?@2� @22 ��� @2?���

���� � �

���@?� @?2 ��� @??

4FFFD

@**Lh@ _i|� '?S

R'�

E���Rn� @�R _i| E��R�c _L�i ��R i *@ tL||L4@|h�Ui _� �

L||i?�|@ _@ � i*�4�?@?_L *@ �@h�}@ i *@ R�it�4@ UL*L??@� �?U�i ^�� _i|�| '_i|� i _i|�� ' _i|�_i|��

,5,��W��� �L?t�_ih�@4L *@ 4@|h�Ui 2� 2

� '

�@�� @�2@2� @22

��

2D

Page 26: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

�@*UL*�@4L _i|� '2S

R'�

E���Rn� @�R _i|��R� 5� �@G ��� ' @22c ��2 ' @2�c

Tih U��

_i|� ' E���2@�� _i|��� n E����@�2 _i|��2 ' @��@22 � @�2@2�

E@?@*L}@ @**@ _i�?�3�L?i _@|@ ThiUi_i?|i4i?|i��

2� �L?t�_ih�@4L *@ 4@|h�Ui �� �

� '

3C @�� @�2 @��

@2� @22 @2�@�� @�2 @��

4D�

5� �@G

��� '

�@22 @2�@�2 @��

�c ��2 '

�@2� @2�@�� @��

�c ��� '

�@2� @22@�� @�2

��

_i|� ' @�� _i|��� � @�2 _i|��2 n @�� _i|����

6Lh4�*@ }i?ih@*i Tih *<�?�iht@G t�@ �R^ *@ tL||L4@|h�Ui L||i?�|@ i*�4��?@?_L _@ � *@ R�it�4@ h�}@ i *@ ^�it�4@ UL*L??@� 5� _i�?�tUi UL4T*i4i?|L�@�� _i**<i*i4i?|L @�� _� �G

�@�� ' E����n� _i|����

�**Lh@

E�3���� '�

_i|��@�� E?L|@hi *L tU@4M�LG �� i ?L? ����

�TT*�U@3�L?i @* U@tL 2�2� 5i �� ' E�@��� i *@ 4@|h�Ui _i� UL4T*i4i?|�c

*@ uLh4�*@ ThiUi_i?|i t� h�tUh��i �3� '�

_i|���|� �**Lh@ ti � '

�@ KS _

�c t�

�@

2S

Page 27: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

�@�� ' _c �@22 ' @c �@�2 ' �Sc �@2� ' �Kc

Tih U��

�� '

��@�� �@�2�@2� �@22

�'

�_ �S�K @

�c ��| '

�_ �K�S @

�c

^��?_� �3� '�

_i|�

�_ �K�S @

��

��A���@�� tih�i @?U�i @ U@*UL*@hi �* _i|ih4�?@?|i �t@?_L *@ h�}@ U�i i T�� UL4L_@

EU�Li ^�i**@ UL? T�� 3ih��G

_i|� 'S�

@���@�� � i tUi*|L _@ ?L��

,5,��W��

� '

3C � � 2

� � ff f 2

4D

�@*UL*�@4L �* _i|ih4�?@?|i tUi}*�i?_L *@ �@h�}@c � ' ��

�@�� ' _i|

�� 2� f

�' �2c �@�2 ' � _i|

�� 2� f

�' 2c

�@�� ' _i|

�� �� �

�' f�

_i|� 'S�

@���@�� ' f � E�2� n f � 2 n 2 � f ' f�

+@?}L _� �?@ 4@|h�Ui� 5�@ � �?@ 4@|h�Ui ?�6� 5� U��@4@ 4�?Lhi _� ��?@ tL||L4@|h�Ui ^�@_h@|@ UL?|i?�|@ �? �� w<Lh_�?i _i* 4�?Lhi i *<Lh_�?i_i**@ tL||L4@|h�Ui ^�@_h@|@�

,5,��W��#@|@ *@ 4@|h�Ui

2.

Page 28: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

� '

3C � f 2

� � ff f f

4Dc

� 4�?Lh� tL?LGLh_�?i �G �c fc 2c �(

Lh_�?i 2G

�� f� �

�c

�� 2� f

�c

�f 2� f

�c

�� �f f

�c

�� ff f

�c

�� ff f

�c�

� ff f

�c

�f 2f f

�c

�� 2f f

�(

Lh_�?i �G

3C � f 2

� � ff f f

4D�

5� _�Ui h@?}L _� � i t� �?_�U@ UL? oE��c *<Lh_�?i 4@tt�4L _i� 4�?Lh� _� �UL? _i|ih4�?@?|i 9' f�

,5,��W��

�� �f f

��@ h@?}L �(

�� �2 2

��@ h@?}L �(�

� ff �

��@ h@?}L 2(

�f ff f

��@ h@?}L f(�

� f ff � f

��@ h@?}L 2(

�� f �� � �

��@ h@?}L 2�

����@4i?|i oE�� 9 4�? i?c6j�

�hLTh�i|@ itti?3�@*i� W* h@?}L _� � i �* ?�4ihL 4@tt�4L _�UL*L??i *�?i@h4i?|i �?_�Ti?_i?|� EL _� h�}�i *�?i@h4i?|i �?_�Ti?_i?|�cTihU�i oE�� ' oE�|���

,5,��W��

� '

3C � f 2

� � ff f f

4D �@ h@?}L 2c TihU�i

2H

Page 29: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

3C �

�f

4D '

2

3C 2

ff

4Dn �

3C f

�f

4D�

AiLhi4@� 5�@ �� ' �� �**Lh@ ti � i �?�ih|�M�*i oE��� ' oE��( ti � i� tL?L �?�ih|�M�*� oE��� ' oE�� ' oE���

�L|@3�L?� 4@|h�U�@*� _i� �i||Lh� i _i**i M@t�� E�L?�i?3�L?i _� ,�?t|i�?�N?@ 4@|h�Ui T�L @?U�i ittihi ULt|�|��|@ _@ L}}i||� T�� }i?ih@*� _i� ?�4ih�hi@*�c M@t|@ U�i t�}*� L}}i||� t�@?L _i�?�|i LTih@3�L?� U�i _@??L ti?tL @ThL_L||� i tL44i( @_ iti4T�Lc UL?t�_ihihi4L 4@|h�U� _� ?�4ih� UL4T*itt� i@?U�i _� �i||Lh��

�L?t�_ih�@4L *@ M@ti e ' E�$e �c�$e 2c ���c�$e ?�c �* �i||Lhi �$� i �* �i||Lhi

UL*L??@ _i**i UL4TL?i?|� _� �$� G � '

3EC

������?

4FD� �**Lh@ �$� '

?S�'�

���$e �� 5� T�L

tUh��ihi

�$� ' e�

i_ @?U�ic Tih �* ThL_L||L tU@*@hiG

�$� � �$� ' �|� ' E��c ���c �?�

3EC

��

����?

4FD ' ���� n ���n �?�?�

A,�+W� #,W 5W5A,�W wW�,�+W

�� N? t�t|i4@ *�?i@hi _� 6 i^�@3�L?� i_ ? �?UL}?�|i i _i**@ uLh4@G

;AAA?AAA=

@��%� n @�2%2 n ���n @�?%? ' K�@2�%� n @22%2 n ���n @2?%? ' K2

���@6�%� n @62%2 n ���n @6?%? ' K6

2b

Page 30: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

_L�i %�c ���c %? 5 U EL F� tL?L *i �?UL}?�|ic K�c ���c K6 5 U EL F� tL?L �|ih4�?� ?L|� i @�� 5 U EL F� tL?L � ULi�U�i?|��

5� T�L @?U�i tUh��ihiG

�% ' Kc

_L�i � i *@ 4@|h�Ui

3EC

@�� ��� @�?���

� � ����

@6� ��� @6?

4FD

% i K tL?L � �i||Lh�

% '

3EC

%����%?

4FD i K '

3EC

K����K6

4FD�

2� �� U��i_�@4L |hi ULti �? t�UUitt�L?iG@� W* t�t|i4@ @44i||i tL*�3�L?�q Et� _�Ui @?U�iG �* t�t|i4@ iUL4T@|�M�*iq�M� "�@?|i tL*�3�L?� U� tL?LqU� "�@*� tL?Lq

�ih h�tTL?_ihi @_ @�c i �|�*i UL?t�_ih@hi *@ 4@|h�Ui ��L||i?�|@ @}}��?�}i?_L @_ � *@ UL*L??@ _i� |ih4�?� ?L|�G

��'

3EC

@�� ��� @�? K����

� � ����

���@6� ��� @6? K6

4FD�

�**Lh@c ti �* t�t|i4@ @44i||i % UL4i tL*�3�L?ic ��L* _�hi U�i *@ UL*L??@

�f

Page 31: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

3EC

K����K6

4FD

i UL4M�?@3�L?i *�?i@hi EUL? ULi�U�i?|� %�� _i**i UL*L??i _i**@ 4@|h�Ui ��W?u@||�c �* t�t|i4@ t� T�L tUh��ihi ?i* 4L_L ti}�i?|iG

%�

3EEEC

@��@2����

@6�

4FFFDn %2

3EEEC

@�2@22���

@62

4FFFDn ���n %?

3EEEC

@�?@2?���

@6?

4FFFD '

3EEEC

K�K2���K6

4FFFD�

"��?_� ?iUitt@h�@4i?|i � i ��_i�L?L @�ihi *L t|ittL h@?}L o Eh�ULh_@hiU�i �* h@?}L i �* ?�4ihL _� UL*L??i *�?i@h4i?|i �?_�Ti?_i?|��� V�Ui�iht@c ti� i ���@??L *L t|ittL h@?}Lc ��L* _�hi U�i *@ UL*L??@ _i� K� i UL4M�?@3�L?i*�?i@hi _i**i @*|hi i ^��?_� �* t�t|i4@ @44i||i tL*�3�L?i� W? @*|hi T@hL*icK 5 W4� E'�44@}�?i _� ��c ^��?_�G

N? t�t|i4@ i h�tL*�M�*i ti i tL*L ti h@?}LE�� ' h@?}LE��� ' o�

�ih h�tTL?_ihi @ M�c UL?t�_ih�@4L Th�4@ �* U@tL L4L}i?iLc U�Li �* U@tL �?U�� K i �* �i||Lhi ?�**LG

K '

3EC

f���f

4FD�

W? ^�it|L U@tLc L���@4i?|ic h@?}LE�� ' h@?}LE��� ' oc Tih U��GN? t�t|i4@ L4L}i?iL i ti4Thi h�tL*�M�*i E% ' f i ti4Thi tL*�3�L?i��W* t�t|i4@ L4L}i?iL i�% ' fc U�Li % 5 !ih�c ^��?_�c t�UUL4i *<@TT*�U@3�L?i

*�?i@hi @ttLU�@|@ @_ � �? �?@ ^�@*t�@t� M@ti i 8 G U? $ U6c �* |iLhi4@ _i**i_�4i?t�L?� U� _�Ui U�i ? ' _�4!ih8 n _�4 W48 � �@

_�4!ih8 ' E_�4i?t�L?i _i**L tT@3�L _i**i tL*�3�L?� _i* t�t|i4@L4L}i?iL� ' ?� o ETihU�i h@?}LE�� ' _�4 W48 ��

5� �t@ _�hi U�i �* t�t|i4@ �@ 4?3o tL*�3�L?��

�i* U@tL ?L? L4L}i?iLc M@t|@ Lttih�@hi U�ic ti % i �?@ tL*�3�L?i ^�@*t�@t�i %�i �?@ tL*�3�L?i _i* t�t|i4@ L4L}i?iLc @**Lh@

��

Page 32: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

%��' %n %�

i @?ULh@ tL*�3�L?i� W?u@||�

�%��' �%n�%�' �% ' Kc

TihU�i �%�' f� V�Ui�iht@c ti % i %��tL?L _�i tL*�3�L?� _�t|�?|i _i* t�t|i4@?L? L4L}i?iLc @**Lh@

�% ' �%��' Kc

U�Li �E%� %��� ' fc U�Li

%� %��5 !ih� ' itL*�3�L?� _i* t�t|i4@ L4L}i?iLj�

"��?_�c ti �? t�t|i4@ _� 6 i^�@3�L?� i_ ? �?UL}?�|i i UL4T@|�M�*ii _� h@?}L oc @**Lh@ @44i||i 4?3o tL*�3�L?��

�ih h�tTL?_ihi @ U�c ti �* t�t|i4@ i UL4T@|�M�*i i _� h@?}L oc t� TLttL?Li*�4�?@hi 6 � o i i^�@3�L?� i UL?t�_ih@hi UL4i |ih4�?� ?L|� ? � o _i**i�?UL}?�|ic Tih @hh��@hi @_ �? t�t|i4@ ^�@_h@|L _� h@?}L 4@tt�4Lc U�Li @_ �?t�t|i4@ ?Lh4@*ic U�Li �? t�t|i4@ _� o i^�@3�L?� �? o �?UL}?�|ic _i* |�TL

�+ ' _ c

_L�i � i �?@ tL||L4@|h�Ui ^�@_h@|@ o � o _� �c UL? _i|� 9' fc + i �?@UL*L??@ UL?|i?i?|i o _i**i �?UL}?�|i %� i _ i �?@ UL*L??@ UL?|i?i?|i o _i�|ih4�?� ?L|� K� tL44@|� @ UL4M�?@3�L?� *�?i@h� _i**i @*|hi ?� o �?UL}?�|ic *@U�� �?�U@ tL*�3�L?i i + ' �3�_ E�?�U@c TihU�i o � o ' f��

A�||i *i tL*�3�L?� _i* t�t|i4@ Lh�}�?@h�L t� L||i?}L?L ^��?_� |h@||@?_L *i?� o �?UL}?�|i U�i UL4T@�L?L �? _ UL4i T@h@4i|h� �@h�@M�*��

,5,��W��� �L?t�_ih�@4L �* t�t|i4@ _� � i^�@3�L?� i � �?UL}?�|iG

�2

Page 33: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

;?=

%n + ' �+ n 5 ' f%� 5 ' �

�MM�@4LG

� '

3C � � f

f � �� f ��

4D c K '

3C �

f�

4D c ��'

3C � � f �

f � � f� f �� �

4D �

W?L*|hic h@?}LE�� ' 2 ' h@?}LE��� ETihU�i *@ e@UL*L??@ i �}�@*i @**@ �@��W* t�t|i4@ i UL4T@|�M�*i UL? 4�32 ' 4 tL*�3�L?��

�@tt�@4L @* t�t|i4@ ?Lh4@*iG }�@h_@?_L *@ 4@|h�Ui �c t� �i_i U�i �*4�?Lhi

�� �f �

�@ _i| 9' fc Tih U�� �* t�t|i4@

�%n + ' �+ ' �5

i ?Lh4@*i _� h@?}L 4@tt�4LG

�%+

�'

���5

�c

_L�i

� '

�� �f �

�c �3� '

�� ��f �

�c _ '

���5

��

�**Lh@

��

Page 34: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

�%+

�'

�� ��f �

����5

�'

�� n 5�5

��

w@ tL*�3�L?i _i* t�t|i4@ i ^��?_�

E� n 5c�5c 5� �

2� 5�t|i4@ _� � i^�@3�L?� �? e �?UL}?�|iG

;?=

5 n 2| ' �2%n e+ � 25 ' e

2%n e+ � 5 n 2| ' .

5� �@

� '

3C f f � 2

2 e �2 f2 e �� 2

4Dc K '

3C �

e.

4Dc ��'

3C f f � 2 �

2 e �2 f e2 e �� 2 .

4D�

5� �ih��U@ U�i h@?}LE�� ' 2 i TL�U�i K i tL44@ _i**i UL*L??i _� �c @?U�ih@?}LE��� ' 2� W* t�t|i4@ i ^��?_� UL4T@|�M�*i i_ itti?_L ? ' ec6 ' �c o ' 2ct� �@??L 4e32 ' 42 tL*�3�L?�� ,*�4�?�@4L �?@ i^�@3�L?i i UL?t�_ih�@4L 2�?UL}?�|i UL4i |ih4�?� ?L|�� �ttih��@4L U�i

_i|

�f �2 �2

�' �2 9' fc Tih U�� �* t�t|i4@G

�5 ' �� 2|

2%� 25 ' e� e+

i ?Lh4@*i _� h@?}L 4@tt�4L�

�%5

�' _

UL?

�e

Page 35: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

� '

�f �2 �2

�c _ '

��� 2|e� e+

�c �3� ' ��

2

� �2 ���2 f

��

�**Lh@

�%5

�' �3�_ '

�� �

2

� f

���� 2|e� e+

�'

��� 2+ n 2� 2+

�� 2|

��

w@ tL*�3�L?i }i?ih@*i _i* t�t|i4@ i

ED� 2+ � 2|c +c �� 2|c |� �

���wW��~W��W wW�,�+W

N?<@TT*�U@3�L?i ) G U? $ U6 t� _�Ui *�?i@hi ti ;�$� c�$� 5 U?c ;@c K 5 Uct� �@

) E@�$� n K�$� � ' @) E�$� � n K) E�$� �

EUL? *i L���i 4L_��U�i t� |h@||@ �* U@tL UL4T*ittL���L?t�_ih�@4Lc @_ iti4T�Lc �* U@tL ) G U2 $ U

2� w@ ThLTh�i|@ _� *�?i@h�|@U� _�Ui U�i Tih UL?LtUihi ) M@t|@ t@Tihi UL4i LTih@ t�� �i||Lh� _i**@M@ti� W?u@||�c t�TTL?�@4L _� �t@hi *@ t|itt@ M@ti e ' E�$e �c�$e 2� t�@ ?i**<U

2 _�

T@h|i?3@ U�i �? ^�i**L _� @hh��L� 5i �$� ' ���$e �n�2�$e 2 ' E�$e �c�$e 2�

����2

�c

@**Lh@

) E�$� � ' ��) E�$e �� n �2) E�$e 2�c

^��?_� M@t|@ _@hi

) E�$e �� ' @�$e � n K�$e 2c ) E�$e 2� ' S�$e � n _�$e 2�

�**Lh@ ) �@ �?@ h@TThiti?|@3�L?i 4@|h�U�@*iG

�D

Page 36: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

) E�$� � ' E�$e �c�$e 2�

�@ SK _

�����2

�' E@�$e � n K�$e 2c S�$e � n _�$e 2�

����2

�c

U�Li

) E�$� � ' E@�$e � n K�$e 2� �� n ES�$e � n _�$e 2� �2

' E@�� n S�2��$e � n EK�� n _�2��$e 2 '

�@�� n S�2K�� n _�2

�c

L��ihLc �? �?@ M@ti ^�@*t�@t� e ' E�$e �c�$e 2�c

) E�$� � ' e��c

_L�i � i *@ 4@|h�Ui U�i h@TThiti?|@ *@ ) i t� ULt|h��tUiG�� 4i||i?_L �? UL*L??@ � |h@tuLh4@|� _i� �i||Lh� _i**@ M@ti

) E�$e �� '

�@K

�c ) E�$e 2� '

�S_

�, � '

�@ SK _

2� LTT�hi 4i||i?_L �? h�}@ � |h@tuLh4@|� _i**i ULLh_�?@|i _i* �i||Lhi �$�

)

����2

�'

�@�� n S�2K�� n _�2

�, � '

�@ SK _

��

V�Ui�iht@c L}?� 4@|h�Ui _@ �?<@TT*�U@3�L?i *�?i@hi ETih �i_ihi U�i U�L �@ti?tLc t� _i�i T@h*@hi _� |h@tuLh4@3�L?i _� M@t�c UL4i u@hi4L �? ti}��|L��

5i ) G U? $ U6 i *�?i@hic UL?t�_ih�@4L _�i L}}i||� �?|ihitt@?|�G�� �* ?�U*iL _� )G

!ih) 'q�$� 5 U

? |@*� U�i ) E�$� � '�$fr(

2� *<�44@}�?i _� )G

�S

Page 37: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

W4) ' i�$� 5 U6 |@*� U�i it�t|i �$� 5 U? Tih U�� ) E�$� � ' �$� j�

5i !ih) 'q�$frEU�Li ?i* ?�U*iL U<i tL*L �* �i||Lhi ?�**L� i ) G U? $ U?

E^��?_� 6 ' ?�c @**Lh@ ) i M��?��LU@c U�Li 4@?_@ |�||L U? t� |�||L U?c �?4L_L U�i �i||Lh� _�t|�?|� �@??L �? �i||Lh� _�t|�?|� i L}?� �i||Lhi R��i?i� _@�? �i||Lhi� W?u@||�c ti _�i �i||Lh� �$� � i �$� 2 �@??L *@ t|itt@ �44@}�?i

) E�$� �� ' ) E�$� 2� , ) E�$� � ��$� 2� '�$f , �$� � ��$� 2 5 !ih) '

q�$fr

, �$� � ��$� 2 '�$f , �$� � ' �$� 2�

W?L*|hic �? ^�it|L U@tL it�t|i )3� i ^��?_� ;�$� 5 U?c �$� ' ) E)3� E�$� ��cU�Li <�$� 5 U?c�$� ' )3� E�$� � |@*i U�i �$� ' ) E�$� �c U�Li W4) ' U?�

5� _�Ui U�i ) i �? �tL4Lh�t4L i U�i *@ 4@|h�Ui � @ttLU�@|@ @ ) i�?�ih|�M�*iG � ' �� , � ' �3���

B*� �tL4Lh�t4� _@??L � U@4M�@4i?|� _� M@tic U�Li |h@tuLh4@?L�i||Lh� �?_�Ti?_i?|� �? �i||Lh� �?_�Ti?_i?|� �? 4L_L M��?��LUL� 5i @MM�@4L�$e �c ���c�$e ? �?_�Ti?_i?|� i �? ?�4ihL R}��t|L�c U�Li ?c }*� �$e�� ' ) E�$e �� tL?L@?ULh@ �?_�Ti?_i?|� i_ �? ?�4ihL R}��t|L�� "��?_�c ti e i_ e�tL?L _�i M@t�c_@|L �? �i||Lhi �$� c t� T�L tUh��ihi �$� ' e�c 4@ @?U�i �$� ' e���� �**Lh@c tie�' e� EUL? � �tL4Lh�t4Lc ^��?_� �?�ih|�M�*i�c ti}�i

e� ' �$� ' e���' e���, � ' ���, ��' �3���

W?L*|hic ti

)E�$� � ' e�� ' e�����, e�� ' e����3�� , � ' ����3�

, ��' �3��� c

�.

Page 38: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

U�Li ti � i *@ 4@|h�Ui @ttLU�@|@ @* U@4M�@4i?|L _� M@tic *i UL4TL?i?|�_i� �i||Lh� |h@tuLh4@?L UL? �3�c 4i?|hi *i 4@|h�U� _i**i @TT*�U@3�L?� *�?i@h�|h@tuLh4@?L UL4i �3����

�**Lh@ )E�$� � ' e�� ' e�����i L}?� 4@|h�Ui _@ �?<@TT*�U@3�L?i *�?�i@hi E�tt@|@ �?@ M@ti��

��A��_i|��' _i|�c TihU�i _i|��' _i|�3� _i|�_i|� ' _i|� i h@?}LE��� '

h@?}LE��c TihU�i oE�3���� ' oE��� ' oE��c itti?_L � i �3� �?�ih|�M�*��

�*|hi ThLTh�i|@ _i**i @TT*�U@3�L?� *�?i@h�� 5�@ ) G U? $ U6 �?<@TT*�U@3�L?i*�?i@hi�

�� !ih) i �? tL||LtT@3�L _� U?� W?u@||�c ti

�$� �c�$� 2 5 !ih) , ) E�$� �� '�$f i ) E�$� 2� '

�$f

, ) E�$� � n�$� 2� ' ) E�$� �� n ) E�$� 2� '�$f , �$� � n�$� 2 5 !ih)�

W?L*|hic ti �$� � 5 !ih) i b 5 U

) Eb�$� �� ' b) E�$� �� ' b � �$f '�$f , b�$� � 5 !ih)�

2� W4) i �? tL||LtT@3�L _� U6� W?u@||�c ti �$� �c�$� 2 5 W4)c @**Lh@<�$� �c�$� 2 5 U? |@*� U�i �$� � ' ) E�$� �� i �$� 2 ' ) E�$� 2��

�$� � n�$� 2 ' ) E�$� �� n ) E�$� 2� ' ) E�$� � n�$� 2�c

U�Li <�$� ' �$� �n�$� 2 5 U? |@*i U�i ) E�$� � ' �$� �n�$� 2c U�Li �$� �n�$� 2 5W4)� W?L*|hic ti �$� � 5 W4) i b 5 U

b�$� � ' b) E�$� �� ' ) Eb�$� �� , b�$� � 5 W4)�

�� AiLhi4@ _i**@ _�4i?t�L?i� 5i ) G U? $ U6 i �?<@TT*�U@3�L?i

*�?i@hi

�H

Page 39: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

_�4!ih)n _�4W4) ' ? �

W?L*|hic ti � i *@ 4@|h�Ui @ttLU�@|@ @ )

_�4W4) ' oE�� �

w@ uLh4�*@ �@ ti?tLc TihU�i @MM�@4L ��t|L U�i ?L? _�Ti?_i _@**@ M@titUi*|@ Tih tUh��ihi �� W?L*|hic t�UUL4i oE�� i �* ?�4ihL _� UL*L??i �?_�Ti?�_i?|� i *i UL*L??i _� � tL?L � |h@tuLh4@|� _i� �i||Lh� _i**@ M@tic @**Lh@ W4)i }i?ih@|@ _@ ^�it|� �i||Lh�� 5� �i_i _@**@ Th�4@ uLh4�*@ U�i ti ? 6*<@TT*�U@3�L?i ?L? T�L ittihi t�h�i||��@ i U�i ti ? : 6 *<@TT*�U@3�L?i?L? T�L ittihi �?�i||��@�

,5,��W��

W?U� UL? *@ M@ti U@?L?�U@

3C �

ff

4Dc

3C f

�f

4Dc

3C f

f�

4Dc UL?t�_ih�@4L *<@TT*�U@3�L?i

)G

) E�$e �� ' �$e � n�$e 2

) E�$e 2� ' �$e � n�$e �

) E�$e �� ' �$e 2 ��$e ��

w@ 4@|h�Ui @ttLU�@|@ i � '

3C � � f

� f �f � ��

4D� AhL��@4L

!ih) '

;?=3C %

+5

4D G �

3C %

+5

4D '

3C f

ff

4D<@>

3C � � f

� f �f � ��

4D

3C %

+5

4D '

3C f

ff

4D /

;?=

%n + ' f%n 5 ' f+ � 5 ' f

,�

+ ' �55 ' �%

�b

Page 40: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

U�Li �$� 5 !ih) / i _i* |�TL

3C %

�%�%

4D ' %

3C �

����

4D�

N?@ M@ti _� !ih) i

3C �

����

4D , _�4!ih) ' ��

AhL��@4L

W4) '

;?=3C %

+5

4D G �

3C %�

+�5�

4D '

3C %

+5

4D<@>�

W?|@?|Lc t� �i_i U�i _i|� ' fc ^��?_� oE�� ' 2� �ihU��@4L Lh@ �?@ M@tiTih W4)� 5� �@

3C � � f

� f �f � ��

4D

3C %�

+�5�

4D '

3C %

+5

4D /

;?=

%�n +�' %%�n 5�' ++�� 5�' 5

, %� + ' 5c

U�Li

3C %

+5

4D 5 W4) ti i tL*L ti i _i* |�TL

3C %

+%� +

4D ' %

3C �

f�

4Dn +

3C f

���

4D�

N?@ M@ti Tih W4) i @**Lh@

3C �

f�

4Dc

3C f

���

4D i

_�4W4) ' 2 ' oE���

ef

Page 41: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

5� TL|i�@ Lttih�@hi U�ic _@**@ _i�?�3�L?i _� )c

) E�$e �� ' ) E�$e ��� ) E�$e 2�c

^��?_� � �i||Lh� U�i t|@??L �? W4) tL?L _i* |�TL

) E�$� � ' @) E�$e �� n K) E�$e 2��

"��?_�

) E�$e �� '

3C � � f

� f �f � ��

4D

3C �

ff

4D '

3C �

�f

4Dc

) E�$e 2� '

3C � � f

� f �f � ��

4D

3C f

�f

4D '

3C �

f�

4D

tL?L �?@ M@ti _� W4)�

�NA�V�w�+W , �NA�V,AA�+W�

5�@ � �?@ 4@|h�Ui ^�@_h@|@ _� Lh_�?i ? hi@*i EL UL4T*itt@� i � �? �i||LhiUL*L??@�

5i �� ' b� UL? b 5 U EL F� i � 9' fc @**Lh@ b i _i||L @�|L�@*Lhi _� �i � i *<@�|L�i||Lhi ULhh�tTL?_i?|i @ b� 5� �?_�U@ UL? Tb *<@�|LtT@3�L _� bcU�Li

Tb ' i� G �� ' b�j ^ ifj�

Tb �?U*�_i |�||� }*� @�|L�i||Lh� i_ @?U�i �* �i||Lhi ?�**L i_ i �? tL||LtT@3�L_� U? EL _� F?��

�55,+V�~W��W E�4TLh|@?|�-���� 5i b i �? @�|L�@*Lhi _� � UL? @�|L�i||Lhi �c @**Lh@ b6 i �? @�|L�@*Lhi

_� �6 UL? }*� t|itt� @�|L�i||Lh�c Tih L}?� 6 �?|ihL TLt�|��L� W?u@||�c Tih6 ' 2c

e�

Page 42: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

�2� ' ��� ' �b� ' b�� ' bb� ' b2��

�**L t|ittL 4L_L t� ThLUi_i Tih 6 }i?ih�UL� 5i � i �?�ih|�M�*ic *L t|ittL�@*i Tih L}?� 6 �?|ihL ?i}@|��L�

2� ��|LtT@3� _� @�|L�@*Lh� _�t|�?|� �@??L �? UL4�?i tL*L �* �i||Lhi ?�**LcU�Li @�|L�i||Lh� _� @�|L�@*Lh� _�t|�?|� tL?L �?_�Ti?_i?|�� W?u@||�c ti

� 5 Tb i � 5 T> , �� ' b� i �� ' >� , Eb� >� � ' f

4@ b 9' > , � ' f�

�L4i t� |hL�@?L @�|L�@*Lh� i @�|L�i||Lh���� ' b� UL? � 9' f t� T�L h�tUh��ihi UL4i E�� bU� � ' fc U�i i �? t�t|i4@

*�?i@hi L4L}i?iL U�i �@ tL*�3�L?i ?L? M@?@*i � 9' f ti i tL*L ti

_i|E�� bU� ' f �

"�it|<�*|�4@ i *<i^�@3�L?i _i}*� @�|L�@*Lh�� E��L @UU@_ihi U�i �?@ LT�� tL*�3�L?� t�@?L UL4T*itti( �? ^�it|L U@tL ?L? tL?L @�|L�@*Lh� ti t� UL?�t�_ih@?L 4@|h�U� hi@*� U�i LTih@?L t� U?� W?�iUic |�||i *i 4@|h�U� UL4T*ittiU�i LTih@?L t� F? �@??L @�|L�@*Lh���

W?_�U��@4L UL?

� Eb� ' _i|E�� bU��

� Eb� i _i||L TL*�?L4�L U@h@||ih�t|�UL _� � i_ �@ }h@_L ?� 5� T�L�ih��U@hi U�i �* ULi�U�i?|i _� b? i E���?c 4i?|hi �* |ih4�?i ?L|L i _i|�E�?u@||�c �* |ih4�?i ?L|L _� �? TL*�?L4�L i � Ef� ic ?i* U@tL _i* TL*�?L4�LU@h@||ih�t|�ULc � Ef� ' _i|E�� fU� ' _i|���

#W�B���wW~~�~W��, #,ww, ��A+W�W�

e2

Page 43: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

N?@ 4@|h�Ui t� _�Ui _�@}L?@*�33@M�*i ti it�t|i �?@ M@ti u@||@ _@ @�|L�i|�|Lh�� W? ^�it|@ M@ti � i tUh�||@ �? uLh4@ _�@}L?@*i UL? }*� @�|L�@*Lh� t�**@_�@}L?@*i�

�<i �? �4TLh|@?|i |iLhi4@ U�i _�Ui U�i *i 4@|h�U� � hi@*� i t�44i|h�U�icU�Li |@*� U�i � ' �|c tL?L _�@}L?@*�33@M�*� EL���@4i?|i ?L? i LMM*�}@|Lh�LU�i �?@ 4@|h�Ui t�@ t�44i|h�U@ Tih ittihi _�@}L?@*�33@M�*i�� �i* U@tL UL4T�

*ittLc *@ UL?_�3�L?i _� t�44i|h�@ i � ' �|c _L�i �

|i uLh4@|@ _@� UL4T*itt�

UL?��}@|� _i}*� i*i4i?|� _� �|��ih �?@ 4@|h�Ui hi@*i 2� 2 i L���LG�

@ KS _

�i t�44i|h�U@ ti K ' S� AhL��@4L }*� @�|L�@*Lh�G _i|E��bU� ' f

_��i?|@

_i|

�@� b KK _� b

�' f / E@� b� E_� b�� K2 ' f

/ @_n b2 � E@n _�b� K2 ' f / b2 � E@n _�b� K2 n @_ ' f�

{ ' E@n _�2 n eK2 � e@_ ' @2 n _2 n eK2 � e@_' E@� _�2 n eK2 A f�

5i { ' fc @**Lh@ K ' f i *@ 4@|h�Ui i }�@ _�@}L?@*i� 5i �?�iUi }*� @�|L�@*Lh�tL?L _��iht�c }*� @�|L�i||Lh� uLh4@?L �?@ M@tic TihU�i tL?L �?_�Ti?_i?|��

5i � i _�@}L?@*�33@M�*ic ��L* _�hi U�i U@4M�@?_L M@tic U�Li T@tt@?_L @_�?@ M@ti u@||@ _� @�|L�i||Lh�c _��i?|@ ��' �3��� c UL? ��_�@}L?@*i� w@uLh4�*@ ?L? i �?�U@c ?i* ti?tL U�i ^�@*t�@t� 4@|h�Ui� U�i �@ UL4i UL*L??i *iUL4TL?i?|� _� �?@ M@ti _� @�|L�i||Lh�c |h@tuLh4@ � �? �?@ 4@|h�Ui _�@}L?@*i�

���#W~W��W #W #W�B���wW~~��WwWA��

5�@ _ Eb� *@4L*|iT*�U�|@ @*}iMh�U@ _i**<@�|L�@*Lhi b EU�Li *@ 4L*|iT*�U�|@_� b UL4i h@_�Ui _i* TL*�?L4�L U@h@||ih�t|�UL� i 6 Eb� *@ 4L*|iT*�U�|@ }i�L4i|h�U@ _� b EU�Li *@ _�4i?t�L?i _i**<@�|LtT@3�L Tb�� 5� �@ ti4Thi � 9

6 Eb� 9 _ Eb�� �ih U��c ?i* U@tL UL4T*ittL EU�Li ?i* U@tL _� 4@|h�U� LTih@?|�t� �i||Lh� �? F?�c t�UUL4i �? TL*�?L4�L _� }h@_L ? �@ ti4Thi ? h@_�U�c t� �@ti4Thi

e�

Page 44: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

S_ Eb� ' ?c

_L�iS

i *@ tL44@ t� |�||� }*� @�|L�@*Lh�� �**Lh@c ti Tih L}?� b t� �@6 Eb� ' _ Eb�c U�Li }*� @�|L�@*Lh� tL?L |�||� hi}L*@h�c i U��@hL U�i *@ 4@|h�Uii _�@}L?@*�33@M�*ic TihU�i U� tL?L @MM@t|@?3@ @�|L�i||Lh� �?_�Ti?_i?|� _@uLh4@hi �?@ M@ti

S6 Eb� ' ?�

�i* U@tL hi@*ic ^�it|L ?L? M@t|@c TihU�i ?L? i _i||L U�i U� t�@?L ?h@_�U� hi@*� i ^��?_� ?L? i }@h@?|�|L U�i

S_ Eb� ' ?c ^��?_� ?i44i?L U�iS

6 Eb� ' ?� �**Lh@c �?@ 4@|h�Ui hi@*i i _�@}L?@*�33@M�*i ti _ Eb� ' 6 Eb�Tih L}?� b i

S_ Eb� ' ?�

��5W ��+AW��w�+W�5i �?@ 4@|h�Ui UL4T*itt@ �@ @�|L�@*Lh� |�||� _�t|�?|� i _�@}L?@*�33@M�*i�

W?u@||�c �? ^�it|L U@tLc _ Eb� ' � Tih L}?� b iS

_ Eb� ' ? i ^��?_� _i�iittihi @?U�i 6 Eb� ' �c ^��?_� t�@4L ?i* U@tL _� @�|L�@*Lh� hi}L*@h�� N?@4@|h�Ui hi@*i U�i �@ @�|L�@*Lh� |�||� _�t|�?|� i _�@}L?@*�33@M�*i tic �?L*|hicS

_ Eb� ' ?�N?@ 4@|h�Ui _i**@ uLh4@

� '

3EEEC

�� f ��� ff �2 ��� f���

���� � �

���f f ��� �r

4FFFDc

_L�i��c�2c ���c�r tL?L 4@|h�U� ^�@_h@|ic i _i||@4@|h�Ui @ M*LUU��� 5�T�L _�4Lt|h@hi U�i �* t�L _i|ih4�?@?|i i �}�@*i @* ThL_L||L _i� _i|ih4�?@?|�_i� �@h� M*LUU�� ��c�2c ���c�r� W?L*|hic Tih *L t|�_�L _� @�|L�@*Lh�c @�|LtT@3�i _�@}L?@*�33@M�*�|@ t� TLttL?L UL?t�_ih@hi � �@h� M*LUU�� tiT@h@|@4i?|i�

�55,+V�~W��,�5i �?@ 4@|h�Ui � i _�@}L?@*�33@M�*i |�||i *i t�i TL|i?3i �?|ihi �6 *L tL?L�

5i � i �?�ih|�M�*i i _�@}L?@*�33@M�*ic @?U�i *i t�i TL|i?3i �?|ihi ?i}@|��i *LtL?L EM@t|@ h�ULh_@hi U�i � i �6 �@??L }*� t|itt� @�|L�i||Lh���

,5,��W��� AhL�@hi �?@ 4@|h�Ui � U�i _�@}L?@*�33@

� '�

2

�� ���� �

�'

3EC

2��

2

��

2

2

4FD

ee

Page 45: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

#i|ih4�?�@4L }*� @�|L�@*Lh�G

_i|E�� bU� '

��

2� b

���

2� b

�� �

e(

_i|E�� bU� ' f / b2 � �bnb

e� �

e' f / b2 � �bn 2 ' f

wi tL*�3�L?� tL?L b� ' � i b2 ' 2��ih b� ' �c t� �@

3EC

2� � ��

2

��

2

2� �

4FD�

%+

�'

�ff

�/

;A?A=

2%� �

2+ ' f

��

2%n

2+ ' f

, % ' +�

�**Lh@

�%%

�i *<@�|LtT@3�L hi*@|��L @ b� i �?@ M@ti ic @_ iti4T�Lc �� '�

��

��

�ih b2 ' 2c t� �@

3EC

2� 2 ��

2

��

2

2� 2

4FD�

%+

�'

�ff

�/

;A?A=

��

2%� �

2+ ' f

��

2%� �

2+ ' f

, % ' �+�

�**Lh@ *<@�|LtT@3�L T2 i ULt|�|��|L _@ �i||Lh� _i* |�TL

�%�%

�� N?@ M@ti

ic @_ iti4T�Lc �2 '

����

��

�h@ TLtt�@4L tUh��ihi � c 4i||i?_L �? UL*L??@ �� i �2G

eD

Page 46: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

� '

�� �� ��

�c �3� '

2

�� �� ��

�c

i t� �@

��' �3��� '

3EC

2

2�

2��

2

4FD

3EC

2��

2

��

2

2

4FD�

� �� ��

�'

�� ff 2

��

2� W? U� UL?t�_ih�@4L *<@TT*�U@3�L?i ) G U� $ U� |@*i U�i

) E�$e �� ' �$e 2() E�$e 2� ' �$e �() E�$e �� ' �$e ��

w@ 4@|h�Ui @ttLU�@|@ @ ) i � '

3C f � f

� f ff f �

4D�

� Eb� ' _i|E�� bU� ' _i|

3C �b � f

� �b ff f �� b

4D ' E�� b� Eb2 � ���

� UL? 4L*|iT*�U�|@ 2(

� Eb� ' f / b ')

�� UL? 4L*|iT*�U�|@ �

��|LtT@3�G T� ' i� G �� ' �j( T2 ' i� G �� ' ��j

�� ' � / E�� U�� ' f /3C �� � f

� �� ff f f

4D

3C %

+5

4D '

3C f

ff

4D

eS

Page 47: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

, oE�� ' �c ^��?_� i*�4�?�@4L _�i i^�@3�L?� i �tt�@4L _�i T@h@4i|h��T� i @**Lh@ �* T�@?L _� i^�@3�L?i % � + ' fc U�Li % ' + i � 5 T� i _i* |�TL

� '

3C %

%5

4D� N?@ M@ti _� T� ic @_ iti4T�Lc �� '

3C �

�f

4Dc �2 '

3C f

f�

4D�

�� ' �� / E�n U�� ' f /3C � � f

� � ff f 2

4D

3C %

+5

4D '

3C f

ff

4D

, oE�� ' 2c ^��?_� i*�4�?�@4L �?<i^�@3�L?i i �tt�@4L �? T@h@4i|hL�

T2 i @**Lh@ *@ hi||@ _� i^�@3�L?�

�%n + ' f5 ' f

i �?@ M@ti i _@|@ _@ �� '3C �

��f

4D�

w@ 4@|h�Ui _�@}L?@*�33@?|i ic @_ iti4T�Lc

� '

3C � f �

� f ��f � f

4D

i t� �@ ��' �3��� '

3C � f f

f � ff f ��

4D�

�� w<@TT*�U@3�L?i *�?i@hi s G U� $ U� _@|@ _@

sE�$e �� ' ��$e 2(sE�$e 2� ' �$e �(sE�$e �� ' �$e �c

?L? i _�@}L?@*�33@M�*i� W?u@||�c *@ 4@|h�Ui @ttLU�@|@ i

e.

Page 48: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

� '

3C f � f

�� f ff f �

4D

i �* TL*�?L?�L U@h@||ih�t|�UL i � Eb� ' E�� b��b2 n �

�� N? tL*L @�|L�@*Lhi

i hi@*i b� ' �� �**Lh@ T� �@ _�4i?t�L?i � i �? �i||Lhi ?L? M@t|@ Tih uLh4@hi�?@ M@ti �? U

�� � i �?�iUi _�@}L?@*�33@M�*i t� Fc U�Li UL4i i?_L4Lh�t4L_� F�c TihU�i �? F �@ |hi @�|L�@*Lh� _�t|�?|�G b� ' �c b2 ' �c b� ' �� i �ULhh�tTL?_i?|� @�|LtT@3� �@??L |�||� _�4i?t�L?i ��

e� w@ 4@|h�Ui

� '

3EEEEC

� � f f ff S f f ff f 2 2 2f f 2 2 2f f 2 2 2

4FFFFD

i @ M*LUU��G �� '

�� �f S

�c�2 '

3C 2 2 2

2 2 22 2 2

4D� W* Th�4L M*LUUL �@

@�|L�@*Lh� � i S E*@ �ih��U@ i �44i_�@|@�� W* tiUL?_L M*LUUL �@ h@?}L �c ^��?_�*@ _�4i?t�L?i _i* ?�U*iL i 2 E_�4!ih�2 ' �� oE�2�� i f i �? @�|L�@*Lhi _�4L*|iT*�U�|@ }iL4i|h�U@ 2 E�?u@||�c _�4!ih�2 ' _�4!ihE�2 � fU� ' _Ef���W?L*|hic h��tU�@4L @ |hL�@hi t�M�|L �? @*|hL @�|L�@*Lhic Lttih�@?_L U�i

3C 2 2 2

2 2 22 2 2

4D

3C �

��

4D '

3C S

SS

4D ' S

3C �

��

4Dc

Tih U�� S i �? @�|L�@*Lhi� w<@�|L�@*Lhi � �@ 4L*|iT*�U�|@ @*}iMh�U@ i }iL�4i|h�U@ �}�@*i @ �� �ih ^�@?|L h�}�@h_@ *<@�|L�@*Lhi Sc ^�it|L �@ 4L*|iT*�U�|@@*}iMh�U@ �}�@*i @ 2c 4i?|hi t� _i�i |hL�@hi *@ t�@ 4L*|iT*�U�|@ }iL4i|h�U@G

3EEEEC

� � f f ff S f f ff f 2 2 2f f 2 2 2f f 2 2 2

4FFFFD

3EEEEC

%+5��

4FFFFD '

3EEEEC

%n +S+

25 n 2�n 2�25 n 2�n 2�25 n 2�n 2�

4FFFFD '

3EEEEC

S%S+S5S�S�

4FFFFD

eH

Page 49: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

,

;AAAAA?AAAAA=

% '|

D+ ' |5 ' o� ' o� ' o

,

3EEEEC

%+5��

4FFFFD '

D|

3EEEEC

�Dfff

4FFFFDn o

3EEEEC

ff���

4FFFFD c

_@ U�� �?@ M@ti Tih *<@�|LtT@3�L _i**<@�|L�@*Lhi S iG

;AAAA?AAAA=

3EEEEC

�Dfff

4FFFFD c

3EEEEC

ff���

4FFFFD

<AAAA@AAAA>c

^��?_� S i hi}L*@hi� W?�?ic f �@ 4L*|iT*�U�|@ }iL4i|h�U@ �}�@*i @ 2c UL4i_i||L tLTh@� �?U�i *@ 4L*|iT*�U�|@ @*}iMh�U@ i �}�@*i @ 2c TihU�i *@ 4@|h�Ui� i _� Lh_�?i Dc TihU�L *@ tL44@ _i**i 4L*|iT*�U�|@ @*}iMh�U�i _i�i _@hi DG6Ef� ' D � 6ES� �6E�� ' D � 2 � � ' 2� �L�U�i |�||� }*� @�|L�@*Lh� tL?Lhi}L*@h�c *@ 4@|h�Ui � i _�@}L?@*�33@M�*i�

A,�+,�� #W ��vw,v�O��WwA���

5�@ � �?@ 4@|h�Ui� �MM�@4L ��t|L U�i � E%� ' _i|E� � %U� i �? TL*��?L4�L _� }h@_L ? ?i**@ �@h�@M�*i % i }*� 3ih� _� � E%� tL?L }*� @�|L�@*Lh� _� ���L?t�_ih�@4L �* ti}�i?|i �4TLh|@?|iG

AiLhi4@ _� �@)*i)�O@4�*|L?� �}?� 4@|h�Ui i h@_�Ui _i* t�L TL*��?L4�L U@h@||ih�t|�UL�

,5,��W��5�@ _@|@ *@ 4@|h�Ui

� '

�� �� �

eb

Page 50: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

�L*�?L4�L U@h@||ih�t|�ULG � E%� ' _i|E� � %U� ' _i|

��� % �� �� %

�U�Li � E%� ' %2 � �%� %n �� � ' %2 � e%n 2�

W* |iLhi4@ _�Ui U�ic ti @* TLt|L _� % 4i||�@4L � i @* TLt|L _� � 4i||�@4L*<�_i?|�|@ Uc L||i?�@4L *@ 4@|h�Ui ?�**@c U�Li

� E�� ' �2 � e�n 2U ' �c

UL? � 4@|h�Ui UL? |�||� f� W?u@||�

�2 '

�� �� �

��� �� �

�'

�2 ee �f

�e� '

� �e �e�e ��2

2U '

�2 ff 2

�2 � e�n 2U '

�f ff f

��

�ih *i 4@|h�U� 2� 2 i ti4Thi

� E%� ' %2 � %Ao�n _i|�c

_L�i Ao� i *@ |h@UU�@ _� �c U�Li *@ tL44@ _i}*� i*i4i?|� t�**@ _�@}L?@*i��L?ti}�i?3@ �?|ihitt@?|i _i* |iLhi4@ _� �@)*i)�O@4�*|L? i �* U@*UL*L

_i**@ 4@|h�Ui �?�iht@ Eti _i|� 9' f��5�@ � E%� ' E���?%? n ���n &�%n &f �* TL*�?L4�L U@h@||ih�t|�UL�E5� h�ULh_� U�i �* ULi�U�i?|i _� %? i ti4Thi E���? i �* |ih4�?i ?L|Lc U�Li

�* |ih4�?i _� TL|i?3@ fc i ti4Thi &f ' _i|���

� E�� ' � / E���?�? n ���n &��n _i|� � U ' �/ � dE���?�?3� n &?3��

?32 n ���n &�Uo ' � _i|� � U/ � dE���?�?3� n &?3��

?32 n ���n &�Uo ' � _i|� � ��3�

_@ U��

Df

Page 51: +WˇOW W 5NW N ,+W ˇ ˘w,55W , ˘ wW W W? U @4L UL? U * …people.unipmn.it/catenacc/geo/algebralineare.pdf · ˝ tL44@GE@cK˝nESc_˝’E@nKcS n_˝( ˝ ThL_L||LGE@cK˝ ESc_˝’E@S

�3� ' � �

_i|�dE���?�?3� n &?3��

?32 n ���n &�Uo

,5,��W��#@|@

� '

3C � � 2

f 2 2f � �

4Dc

t� �@ � E%� ' �%� n S%2 � b%n ec ^��?_�

�3� ' ��

eE��2 n S�� bU� '

eE�2 � S�n bU� '

e

3C e �� �2

f � �2f �� �2

4D

D�