Encrypted Document with References to University Topics and Academic Coursework - Prof. J. | Study notes Linear Algebra

    

!"$#% &'#()*+#

,.-0/21436587:9<;>=@?BA@=;698/$C8/EDEFHGJIKFH143ML436FH5ONQP:RSTU3V1=WGX=YLZ?Q3M[ \]=^D_LZFH?Q36`=L436FH5ba$-8FH1c/dDEFHGbe

I89<Lf=YL436FH53V1b=g1hLZ/iIj365kGX=@5:lm=Y;67HF:?Q3nLZ-0G 1Eoqp)rs9t=Y;M;nlu36GJIvF:?QLf=^5HLw3V1xLZ-8/981c/XF@\LZ-8/

P:RSy\zF:?DEFH58DE/EI<LZ9t=Y;{I898?4IKFH1c/i1Eo}|=@5:lXI8?4F:~<;6/iGJ1F@\;M3658/=@?.=Y;67H/E~8?Z=wD=@5~K/W~K/*L4LZ/E?

9858C0/i?41hLZF<F<C3M\ad/t?414L=@1cXLZ-8/rs98/E14L436FH5<a$-t=YL.3M\ad/WLf=^@/WLZ-8/bP:RS

 BdU8cw8K}Zd

,.-0/P:RSy361.GJF@L43MA@=YLZ/iC~:lLZ-8/\zF@;M;VFia.36587J7H/EF:G /_LZ?Q36D\]=@D_Li

¡£¢b¤>¥ ¦§:¢¨4©wª«¡£¢¬<®¤zª°¯]±t¡<¢E²4¢w¬<®v³:¢E²¦@®´xµ·¶¸¥ ¦:ªz²¹¤nºX¤>¯¦ ¡H´f±8¢E²4¢»>»¼¤½±v¯_¢¾

,.-0/OP:RS¿3V1b=@I8I£;n36D=@~<;6/bLZF~vF^LZ-k?4/=Y;=^58CuDEF:GJI<;V/f[jGX=YLZ?Q3VDE/E1ioqÀ$Fad/_AH/E?*U365kC8/_e

1ZDE?Q36~<3V507LZ-8/J7:/iF:GJ/_LZ?Q3VDw3V5:LZ/E?4I8?4/_Lf=YL43VF:5ad/X=@1Z1c98GJ/=^1W981Z9t=Y;LZ-t=YLLZ-8/JGX=LZ?Q3Á[3V1

?4/=;]o

,.-8/LZ/E?4GÃÂc-:l0IK/E?4/_;n;M3VI01Z/EÄ GX=ElÅ~v/W985£\]=@Gb3M;n3>=@?*~09<L.LZ-<3613V1BÆ49814LdLZ-8/xµeÇC£3VGJ/E5<e

1436FH5=;.7:/E58/E?Z=;M36`Y=YL436FH5ÈF@\W=^5É/_;M;n36I81Z/@omÊm/ÅGË=*lmC8/_t58/=-:l<Iv/E?4/_;M;n36I81Z/q365mÌ ÍdÎ=@1

LZ-8/1Z90?Q\]=@DE/FH~£Lf=3658/EC~:l14LZ?4/_LZDf-<36587ÈLZ-8/985<3MLg1cI8-8/E?4/q3V5Ì ÍdÎ~:l1ZF:GJ/g\]=^D_LZFH?41

ÏvÐ¹ÑiÒEÒEÒÑ¹Ï

IKF:1Z1436~<;nl`E/E?4F.3V5Ó1ZF:G /F:?QLZ-8F:7HF:5t=Y;C<36?4/ED_L43VF:581OÔ

Ð*ÑEÒEÒEÒiÑ

Î Õ

Ì Í+Îo

Ö8F:?JDEFH5:AH/E5<36/E58DE/@;6/_LX981bLf=@^/gLZ-0/gÔ0×.LZFk~K/q985<3ML A:/ED*LZF:?41io,.-0/OA:/iD_LZF:?41gØ

×>Ô8×ÇÙ

=^?4/LZ-0/±K²f¤>®KÚE¤½±8¦H»U¯*¢E¥b¤z¦Yº:¢E¯bF@\}LZ-8/J-:l<Iv/E?4/_;M;n36I81Z/@a.3MLZ-u;6/E587@LZ-01

ÑEÒEÒEÒÑfÏ

obÌÛ\dÜ

-t=^1?Z=^58jÝ£}/f[=@D_L4;MlmÝF^\.LZ-8/Ë;V/E587^LZ-81

×a3n;M;dLZ98?45'F:9<LbLZF~v/Å58FH50`i/E?4F8}=^58CÈ3V5

It=^?QL43VDE9<;>=^?*03n\°µßÞ¸°K=YL.GJF:14L.¸uF^\LZ-8/EGa3n;M;~v/W50FH58`E/E?4F8o

àHá

Encrypted Document with References to University Topics and Academic Coursework - Prof. J., Study notes of Linear Algebra

Related documents

Partial preview of the text

Download Encrypted Document with References to University Topics and Academic Coursework - Prof. J. and more Study notes Linear Algebra in PDF only on Docsity!

! HÐ !#"

Ü

Ï

Ô

Ï ÐÔ Ð

Ü

; }Î2B58F:?361 F=Ë985<3MLf=@?QlkGX=YLZ?Q3M[

F:?QLZ-8FH50FH?4GX=Y;UDEF^;V90G 5 01i FD=@5Å~v/W/f[£LZ/E58C8/ECqLZFX=905<3nLf=^?QlOGX=YLZ?Q3M[

ÌÇ\ F/ 3 61?4/EI<;>=@DE/ECk~:l F365ÓNJ(8o à TZULZ-8/E5 1 / a.3M;n;d-t=*AH/XLZF D¹-t=@507H/JLZF<F8oÖ8F:?WLZ-8/

~:lO`E/E?4FoDEDEF:?4C<3V507@;MlHs;6/_L 1 Ó~v/LZ-0/μ ¶q¸ GX=YLZ?Q3M[qDEF:58143614L436587JF@\ 1 É365OLZ-8/W90I8Iv/E?/

Ü E F 1

,H 2 Ò NJ( ÒK£T

À$/E?4/ F3V1Wμ ¶μ =^58Cu985<3MLf=^?Qls H 3 V1¸'¶ ¸=^58Cu985<3MLf=^?QlsB=^58C 1 y361μ ¶¸ =^58C

Ü F 1 H 2

,.-0/ C=@1Z-0/iC;n3658/E1W3658C<36DY=YLZ/bLZ-8/kÂc143n;6/E5:LZÄgDEF^;V98GJ581xF@@F=^58Cu?4Fia21F^!1LZ-t=YL=^?4/

7 :/iF:GJ/_LZ?Qlso,.-8/JGX=YLZ?Q3M[1ya3n;M;58Fa¿-t=EAH/JÝIKF:143nL43MAH/JC<3>=^7HF:5t=Y;)/E5:LZ?Q3V/E1ia.3MLZ- LZ-8/

ÜF^\;6/E1Z1$LZ-t=^5g\z9£;n;°?Z=@58to Q258/D=^5gLf=^@/ F/ LZF~v/wμ ¶¸a.3MLZ- 1 / F^)C<36GJ/i50143VF:

1 ZFbLZ-t=YL F/ 3 V1μ¶qÝJ=^58C 1 / 3 61.Ý ¶gÝ=^58Cg1hLZ?Q3VD_L4;MlOIvF:143ML43nA:/F:5LZ-8/C£3«=^7HF:5t=Y;Ûo

4 

; Î 1452 

13H 2 NJ( Ò(0T

/       ,      à

; 2 452 3 V1985<3MLf=@?QlH

Õ

ÌQ5u=@C8C£3nL436FH5

5 8F:5<3658DE?4/Y=^1436587JFH?4C0/i?D£LZ-t=YL.361i ÏKÐ Þ Ï " Þ 8 :8P8vÞ Ï Þ

0a2-0/i?4/^ E G365N^ μ Ñ¸TZo

? 2F^LZ/$LZ-t=YLdLZ-8/WC<3>=@7:F:5t=;

ÌÇL}361D_;6/=@?BLZ-t=YLLZ-0/.36GX=@7:/.F^\vLZ-0/2985£3nL1ZI8-0/i?4/+3V5JÌ Í 32 9858C0/i?=WGX=^IXÜ E F13H 2

G 9 014L.~K/x=w-:l<Iv/E?4/_;n;M36I81Z/365XÌ Í+ÎoË,-8/985<3MLf=@?QlGX=^IH 2 I8?4/E1Z/E?QAH/E1dLZ-8/W1cI8-8/E?4/@sLZ-8/

C<3>=@7:F:5t=;GX=YLZ?Q3M[ 1 1hLZ?4/*LZD¹-8/E1bLZ-8/1ZI8-0/i?4/Ë3V5:LZFu=-:l<Iv/E?4/_;M;n36I81Z/q=;M3V7:58/ECja.3MLZ-kLZ-8/

D=@50FH5<36D=;s~t=^14361i^=@58CxLZ-8/t5=;<985<3MLf=^?QlGX=@I F?4F^Lf=LZ/E1F:?{?4/t/ED_LZ1LZ-8/-:l<IK/i?4/_;M;n36I81Z/

  Z<tB   B8

   36 ¢E²f´¥J¦Hª«²f¤nºXÜ

Õ

;)Î14#2k¡£¦@¯X¦¯f¤>®<§^¬8»Á¦^² 6 ¦:» ¬8¢Ë³H¢¹Úf¨^¥2±8¨^¯f¤zª«¤z¨@®

N J(8o (<TE¾"!

Ü ¤>¯d¯$%E¬8¦^²4¢¦^®v³bª«¡£¢ Ï C ¦^²c¢W³^¤>¯*ª«¤>®vÚª&#{ª>¡<¢W» ¢]©_ª{¦@®K³²¹¤n§¡<ª{¯¹¤>®0§^¬8» ¦@²,6¢¹Úª]¨^²f¯ØiÔ CÙ ¦@®K³

Ø !C ÙÅ¦@²4¢x¬<®t¤%E¬8¢» ´w³H¢ª ¢E²f¥b¤>®v¢f³b¬±uª ¨ Ú¹¨@¥$±

¨4©¦_¯¨:» ¬8ª ¢'6¦:» ¬8¢,+ - s¾

.+²4¨¨c©E¾,BFXI8?4FAH/b/f[£3V14LZ/E58DE/bF@\°LZ-8/XP:RSad/361ZF^;«=YLZ/LZ-0/C<36?4/ED_L43VF:5 F@\°LZ-8/w;«=^?47:/i1hL

Ps/_L Ï Ð E /Ü /"o9}l =ÈDEFHGJIt=^D_LZ58/E1Z1O=^?47H90G /E5:Li2LZ-8/E?4/uG981hLg~K/k=jAH/ED_LZFH?

! Ð Õ^ ;^2 a3nLZ-^ /! Ð /" E )g=^58C^ /Ô Ð/" E ÏvÐ a$-8/E?4/gÔ^ Ð E Ü! Ðo ;FH50143VC0/i?X=^5:l

/f[£LZ/i50143VF:581F^\ ! Ð LZF$=@5xFH?QLZ-8F:58F:?4GX=;s~t=^143V1)Ø !DC ÙF@\ ; 2 2=@50CF@\8Ô Ð LZF=@5WF:?QLZ-8F:58FH?4GX=Y;

~t=^14361$ØiÔ CÙF@\ ; Î 0=@50C;6/_L@F Ð =@58CBH Ð C8/E58F^LZ/dLZ-8/985£3nLf=^?Ql GX=YLZ?Q3VDE/E1}a.3MLZ-XDEF@;698GJ

F Ð 2 Ü H Ð E  E 10 Ï Ð^32

Ñ

NJ(

a2-0/i?4/

3 61}=2DEF^;V90G 5 wAH/ED_LZF:?F^\vC<36GJ/i50143VF:5μ $ )@ 2 2 361)=?4FiamA:/iD_LZF:?)F^\tC<36GJ/E581436FH

Ï Ð 12 2

Ï Ð

Þ

Ï

87 2 22 E NÏ

Ð 97 2 22 T

Ð

;: "^66

Ï Ð

Ñ

3 6G I£;nls36587 /  /" Þ NÏ "Ð<7 2 22 TÐ;:^ "oXP:3658DE/ F Ð =^58C H Ð =@?4/J985£3nLf=^?QlH

/  /"@E=/ Ü /"@E Ï Ð1ZFbLZ-<3613VGJI<;M3V/E1^2 E = o

/       ,      K )

Êm?Q3MLZ/W=    I0?4FH7:?Z=@GN 1 Z/E/ 0 {/ED_LZ98?4/ TBa$-<3VD¹-07@3MAH/E5O=w?4/=; à ¶ à GX=LZ?Q3Á[ÅÜ

lHF:98?I8?4F:7:?Z=@G¿LZFLZ-0/GX=YLZ?Q3M[ NK<o  T°=^58CO=Y;V1ZFLZFLZ-8/ à ¶ à GX=LZ?Q36DE/i13V5Op[0/E?4D_3V1Z/ )@o

·Õ

Î

74 Î =^?4/g¬<®¤zª]¦^²f¤z»¼´¢ %E¬<¤ 6 ¦H» ¢E®vªd3M\dÜ E 

 2 \zF:?1ZF:GJ/

; Î 74 Î oÌh1$3ML2LZ?498/bF:?.]=Y;V1c/WLZ-t=YLxÜ¿=@50C

;)Î 1452 g-t=^1 =^5ÉP@RwS Ü E F1,H 2 o PH-8Fa

ÌÍ+Î 14 Î./H

Õ

; }Î2B58F:?361 F=Ë985<3MLf=@?QlkGX=YLZ?Q3M[

F:?QLZ-8FH50FH?4GX=Y;UDEF^;V90G 5 01i FD=@5Å~v/W/f[£LZ/E58C8/ECqLZFX=905<3nLf=^?QlOGX=YLZ?Q3M[

ÌÇ\ F/ 3 61?4/EI<;>=@DE/ECk~:l F365ÓNJ(8o à TZULZ-8/E5 1 / a.3M;n;d-t=*AH/XLZF D¹-t=@507H/JLZF<F8oÖ8F:?WLZ-8/

~:lO`E/E?4FoDEDEF:?4C<3V507@;MlHs;6/_L 1 Ó~v/LZ-0/μ ¶q¸ GX=YLZ?Q3M[qDEF:58143614L436587JF@\ 1 É365OLZ-8/W90I8Iv/E?/

À$/E?4/ F3V1Wμ ¶μ =^58Cu985<3MLf=^?Qls H 3 V1¸'¶ ¸=^58Cu985<3MLf=^?QlsB=^58C 1 y361μ ¶¸ =^58C

,.-0/ C=@1Z-0/iC;n3658/E1W3658C<36DY=YLZ/bLZ-8/kÂc143n;6/E5:LZÄgDEF^;V98GJ581xF@@F=^58Cu?4Fia21F^!1LZ-t=YL=^?4/

7 :/iF:GJ/_LZ?Qlso,.-8/JGX=YLZ?Q3M[1ya3n;M;58Fa¿-t=EAH/JÝIKF:143nL43MAH/JC<3>=^7HF:5t=Y;)/E5:LZ?Q3V/E1ia.3MLZ- LZ-8/

ÜF^\;6/E1Z1$LZ-t=^5g\z9£;n;°?Z=@58to Q258/D=^5gLf=^@/ F/ LZF~v/wμ ¶¸a.3MLZ- 1 / F^)C<36GJ/i50143VF:

1 ZFbLZ-t=YL F/ 3 V1μ¶qÝJ=^58C 1 / 3 61.Ý ¶gÝ=^58Cg1hLZ?Q3VD_L4;MlOIvF:143ML43nA:/F:5LZ-8/C£3«=^7HF:5t=Y;Ûo

4

; Î 1452

13H 2 NJ( Ò(0T

/ , à

; 2 452 3 V1985<3MLf=@?QlH

5 8F:5<3658DE?4/Y=^1436587JFH?4C0/i?D£LZ-t=YL.361i ÏKÐ Þ Ï " Þ 8 :8P8vÞ Ï Þ

0a2-0/i?4/^ E G365N^ μ Ñ¸TZo

? 2F^LZ/$LZ-t=YLdLZ-8/WC<3>=@7:F:5t=;

ÌÇL}361D_;6/=@?BLZ-t=YLLZ-0/.36GX=@7:/.F^\vLZ-0/2985£3nL1ZI8-0/i?4/+3V5JÌ Í 32 9858C0/i?=WGX=^IXÜ E F13H 2

G 9 014L.~K/x=w-:l<Iv/E?4/_;n;M36I81Z/365XÌ Í+ÎoË,-8/985<3MLf=@?QlGX=^IH 2 I8?4/E1Z/E?QAH/E1dLZ-8/W1cI8-8/E?4/@sLZ-8/

C<3>=@7:F:5t=;GX=YLZ?Q3M[ 1 1hLZ?4/*LZD¹-8/E1bLZ-8/1ZI8-0/i?4/Ë3V5:LZFu=-:l<Iv/E?4/_;M;n36I81Z/q=;M3V7:58/ECja.3MLZ-kLZ-8/

D=@50FH5<36D=;s~t=^14361i^=@58CxLZ-8/t5=;<985<3MLf=^?QlGX=@I F?4F^Lf=LZ/E1F:?{?4/t/ED_LZ1LZ-8/-:l<IK/i?4/_;M;n36I81Z/

Z<tB B8

36 ¢E²f´¥J¦Hª«²f¤nºXÜ

;)Î14#2k¡£¦@¯X¦¯f¤>®<§^¬8»Á¦^² 6 ¦:» ¬8¢Ë³H¢¹Úf¨^¥2±8¨^¯f¤zª«¤z¨@®

Ü ¤>¯d¯$%E¬8¦^²4¢¦^®v³bª«¡£¢ Ï C ¦^²c¢W³^¤>¯*ª«¤>®vÚª&#{ª>¡<¢W» ¢]©_ª{¦@®K³²¹¤n§¡<ª{¯¹¤>®0§^¬8» ¦@²,6¢¹Úª]¨^²f¯ØiÔ CÙ ¦@®K³

Ø !C ÙÅ¦@²4¢x¬<®t¤%E¬8¢» ´w³H¢ª ¢E²f¥b¤>®v¢f³b¬±uª ¨ Ú¹¨@¥$±

¨4©¦_¯¨:» ¬8ª ¢'6¦:» ¬8¢,+ - s¾

.+²4¨¨c©E¾,BFXI8?4FAH/b/f[£3V14LZ/E58DE/bF@\°LZ-8/XP:RSad/361ZF^;«=YLZ/LZ-0/C<36?4/ED_L43VF:5 F@\°LZ-8/w;«=^?47:/i1hL

Ps/_L Ï Ð E /Ü /"o9}l =ÈDEFHGJIt=^D_LZ58/E1Z1O=^?47H90G /E5:Li2LZ-8/E?4/uG981hLg~K/k=jAH/ED_LZFH?

! Ð Õ^ ;^2 a3nLZ-^ /! Ð /" E )g=^58C^ /Ô Ð/" E ÏvÐ a$-8/E?4/gÔ^ Ð E Ü! Ðo ;FH50143VC0/i?X=^5:l

/f[£LZ/i50143VF:581F^\ ! Ð LZF$=@5xFH?QLZ-8F:58F:?4GX=;s~t=^143V1)Ø !DC ÙF@\ ; 2 2=@50CF@\8Ô Ð LZF=@5WF:?QLZ-8F:58FH?4GX=Y;

~t=^14361$ØiÔ CÙF@\ ; Î 0=@50C;6/_L@F Ð =@58CBH Ð C8/E58F^LZ/dLZ-8/985£3nLf=^?Ql GX=YLZ?Q3VDE/E1}a.3MLZ-XDEF@;698GJ

F Ð 2 Ü H Ð E E 10 Ï Ð^32

3 61}=2DEF^;V90G 5 wAH/ED_LZF:?F^\vC<36GJ/i50143VF:5μ $ )@ 2 2 361)=?4FiamA:/iD_LZF:?)F^\tC<36GJ/E581436FH

3 6G I£;nls36587 / /" Þ NÏ "Ð<7 2 22 TÐ;:^ "oXP:3658DE/ F Ð =^58C H Ð =@?4/J985£3nLf=^?QlH

/ /"@E=/ Ü /"@E Ï Ð1ZFbLZ-<3613VGJI<;M3V/E1^2 E = o

/ , K )

Êm?Q3MLZ/W= I0?4FH7:?Z=@GN 1 Z/E/ 0 {/ED_LZ98?4/ TBa$-<3VD¹-07@3MAH/E5O=w?4/=; à ¶ à GX=LZ?Q3Á[ÅÜ

lHF:98?I8?4F:7:?Z=@G¿LZFLZ-0/GX=YLZ?Q3M[ NK<o T°=^58CO=Y;V1ZFLZFLZ-8/ à ¶ à GX=LZ?Q36DE/i13V5Op[0/E?4D_3V1Z/ )@o

74 Î =^?4/g¬<®¤zª]¦^²f¤z»¼´¢ %E¬<¤ 6 ¦H» ¢E®vªd3M\dÜ E

2 \zF:?1ZF:GJ/

; Î 74 Î oÌh1$3ML2LZ?498/bF:?.]=Y;V1c/WLZ-t=YLxÜ¿=@50C

;)Î 1452 g-t=^1 =^5ÉP@RwS Ü E F1,H 2 o PH-8Fa

ÌÍ+Î 14 Î./H