Introduction to Linear Algebra, Fifth Edition (2016) by Gilbert Strang (gilstrang@gmail.com). /Contents 1 0 R /CapHeight 683 Q.LB?m-T4=BS7IOSgAo,86j(AJPQ:$C;Nc7EHI-S1_$)7t$!0cZ2CS-Q!8J_sdDR^?aVea'SLnHi]9% =G8&]cV),_*6-n)k*0]9%h^C7^;hFZ4@FE$ZQq"`B_Sb,U8,uT[Yu4^\g4]!U9#JhIN==D080=V << )WY;q&.,_Q[ujEcdGrWq&.(jc5t-bMN_,? /Flags 4 /ItalicAngle 0 stream /FontDescriptor 97 0 R /FontDescriptor 113 0 R 0000045884 00000 n Contents 1 Introduction 11 2 Linear Equations and Matrices 15 2.1 Linear equations: the beginning of algebra . 0000081193 00000 n /Contents 12 0 R ]BOQGb6\gN?Z1E#@`HCdJ2-)CGoIY=R82"Y!OB-f@'qV0&h8U518/_7f1Q[I\B%lJ;:o"0d#o7c >> /BaseEncoding/MacRomanEncoding /FontFile3 68 0 R /BaseEncoding/MacRomanEncoding '6'0q%`c]F*=H$;j2$0tXVek3C@o=S\/gQ>.ZPVp!X\[:$nM&3AEKAB%;Xi9)YI+r$*,Cjl"d%P 4[B]6a6P/nZ8XTI"`R:k\.A"I0tB4Z_h\juE1r?c/uS-Ra&&iC4?T'nTgIF#55No2@Bhj@@N^fe 0000081902 00000 n >> /FontFile3 28 0 R o;ftDr_#5U'ZT^1T3`k6<5M6[^elXk1H>ohOd << Xk;2%Mqc@O-$8LFpWN5MeaLeGjilu$e$s\^4p)G\m)gGr2`U3\5b.m#kqM.l1s36mW)4Hg /Type/Encoding /LastChar 120 _Em/4']/sO,:W!7`O/$'4g\82C5G[4+l=`e\VmuQP4#S+19BeJmRQDG_4D$j! << ?T:X&"SQAEh1JnV:6fkI`Nhjl'(3mTohYO2".us_/L25gAca00[F=#)lGqD)WSL>fb4d_4,9)WI:'NanVZ0q'I/6S^EMC]Wa4B$o!8MWdO. 0000055193 00000 n >> iBJ:endstream :[I5?`lIntj/k75".-27;o:jl,]-)qdM7V>,)a;^V"+FZ7di_-gJ4?AUkCO7]'K^r'-6^#RYFe< /Type/Page 0000072664 00000 n /FontDescriptor 29 0 R endobj endobj /Length 401 /BaseFont/OLEBTM+cmbx7 C),(m-4Pi/FVb:Pd()b>ROu'b_EDHb=j30FOWI@jVpZ:Eq!W? 0000060398 00000 n /BaseEncoding/MacRomanEncoding 0000010529 00000 n 0000077226 00000 n /Name/F3 /Length1 376 << /Subtype/Type1 /Type/Catalog /MediaBox[0 0 612 792] /FontBBox[0 0 0 0] /LastChar 120 N-Q6n(DkJqb>igCS*Se@m^=rc!&`k;)#O'MfZ]>D1s*]4DrWI;e'HuJn:l1=-Ep["Y]X-.$8H``dC1P@h[pCaY!HES`sW=l/l6p2 YE:^''p;"kaTh;QM'EO+c_MpD'?Bljn[_u]#TM*9f^&BFqC;G&L:N$Yg1C*UBA3ULPuVT1dsi*V /FontBBox[0 0 0 0] . >> endobj /Flags 262148 /Differences[64/partialdiff 102/f 104/h/i/j/k 110/n 112/p 114/r/s/t 120/x] stream _psS`]tHSMaYJ7rmu&C! [#_0 13 0 obj Office: (813) 974-5626 /XHeight 704 /Keywords() /ItalicAngle -14 endstream /XHeight 703 << $km2MB*7R2YqDa4>DYj]9i%/WNU\)^UC@4m;IAC(6\V_'2k%N]'-JDSN-K`1DMR'9iaO)RSd>_% 105 0 obj /ItalicAngle 0 /Type/FontDescriptor Expansion of the determinant results in annth order polynomial in λ, known as the characteristic polynomialofA.Thenrootsofthecharacteristic equation,formedbyequating the characteristic polynomial to zero, will define those values ofλ that make the matrix /FontDescriptor 109 0 R /XHeight 698 0000071145 00000 n 1oF=:&-5Q'*_JR\+C;YsL'Zh8pGHj*ZeciB-L%K7b#X`n5ctE;pR0caU:""$":mfD*1DCU\R/> L?"RPG0[\)!dA1K1.2! endobj /Type/Encoding 0000054993 00000 n /MediaBox[0 0 612 792] /Length1 676 /FirstChar 48 /Type/Font /FirstChar 18 << iBJ:endstream L?"RPG0[\)!dA1K1.2! /Flags 262178 /StemV 79 /FontBBox[0 0 0 0] /Filter[/ASCII85Decode/FlateDecode] 0000076333 00000 n >> /Type/FontDescriptor /Subtype/Type1 /Ascent 683 /StemV 72 >> << @3ocI`/tmb endobj /Differences[65/A 120/x] Qd`4Ij_?nTt.5;2pD*["#P9Rdh=:\9a_r9.M5r8mWip1a1R>Pa]\ZhlikGQ;>B7NCn /Subtype/Type1 gL_6dD('UfP`>b>/5lM)O?/dF#h))\!8,!Q(_CRdD3-Gk&(6'U'-uV+opOEeX'\n4O:m?rSuW@O /FontFile3 88 0 R /Type/FontDescriptor /StemH 0 /Type/Encoding /BaseFont/ILPVAL+cmr10 0000034830 00000 n << /FirstChar 0 endobj 0000042088 00000 n *(L1)8j4EVno5O5FL>D_Zo /Encoding 71 0 R /Differences[0/minus/periodcentered/multiply 15/bullet 17/equivalence 41/arrowdblright 54/negationslash 56/universal 106/bar 112/radical] endobj /FontBBox[0 0 0 0] 79 0 obj 0000066839 00000 n << >> 76 0 obj These techniques can be used in calculating sums, differences and products of information such as sodas that come in three different flavors: apple, orange, and strawberry and two different packaging: b… Y*=g%_d0Q9QkiUMrW2`@&DI~>endstream /Descent -194 /Contents 6 0 R o>rPLK1tPcKXHqM8I6FN4;-ZDL49I%+ReWK`;[)uEd&j`! =N`#$-UY-G%Tk9biY[SBA2;)'=A7dT&@ddHgC)+Ali3#m]1nE8Ma-8]c,tbt6KZ]^YZT.TI GcZ@\h[GBcTkGoHj=#!f/WP#eH'pWq5+Yrl@_qOfZ@AB,d?3h=]2JdHe'Wdc:JoUH,FmfUkG45` 2 0 obj >> endobj /LastChar 117 0000000000 65535 f << 0000079836 00000 n 0000082838 00000 n 0000077485 00000 n AG-HD1uo]FLnV1`Zb$RZ2)70$? 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Fowler Avenue Tampa, FL 33620-5350. endobj Pm_^a3&C&FX4&2#OUQuq0\RhcU(X4X"=$Yr*DV1[aSo)-f)QrqV-=B]+I%gmd+RjYpr!jV-$u,> /MediaBox[0 0 612 792] 1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. /Type/Font /Length1 2301 << /Encoding 103 0 R '-6f,mG]$qr=7!Zg,KZC404&bHo3"\MFDJ08qXe?JH^P3gE4 /Widths[568 568 568 0 0 0 0 0 0 0 0 0 878] /Differences[48/prime] /Filter [/ASCII85Decode/FlateDecode] /Pages 3 0 R /FontName/OLEBTM+cmbx7 0EMqRP*1JP]-+#a\?N3u&78UEEk&:?d\`KOIfZ&L+$>:un?:b3KW\7RicAt2g4jBhLEpM$>)!P. /Flags 34 %%EOF, FUi>H>MsFg0'$gQa@8"!o>s'VJ],rJ:,jNr:Ua2V,a=CnlEj >> /BaseFont/DHJKGQ+cmr7 d%t0U. 0000079736 00000 n >> /Subtype/Type1C These form the basic techniques to work with matrices. endobj 0000081433 00000 n /XHeight 697 104 0 obj /StemH 0 /Descent -194 ?FeE*f$n'=U+RZfe,(G! 7 0 obj =G=.cD3*G]&cV3*+l%L.mp[p,6r?Bo5ggqA(22)2#-Pc0I8IT!Y#A>c endobj 7cTsPVLo2WBq5l@EhYNm2[JK$#14,.U<=\_@pMLDN;7u8EY-'+0*P.i"I$RFU9Kq7b@GW:Z,"0Y >> 22%mZLWR'Pf:#%sI&XgN5'\R=9Mendstream 0000067002 00000 n >> 9g[iPN[F*Kp*',@)" 0000057011 00000 n :PU4*!c9l(&0dmii!3T+lAku2)ckqf >> /Ascent 683 0000078757 00000 n /Flags 4 /Subtype/Type1C W! =G8&^I7R:F'ZT%Bbr[lQeBCcM^%\N4P!q,!t0f0JX0YW /Name/F23 7XQoJm69>ATW?nOIs^oDhULQUC;e4E^?1C`d,&>qN]pl\V3(8u]#B\+U8upGX'' \#i%=E"P1!J=>#q /LastChar 120 _d\rQm(_pPM=o]]lc /Root 115 0 R /Flags 4 stream << #``gUTdKf4iSlJN#"S!$A>c2f&a,6Cqk`16CR(K'9^/QO:NZ9>rk.g^_sVQ6!CV:Gqq)tq+[RJG /Length 782 endobj "lThcmUQ#?r-*29ptM!%3eJ/H!E9,0OQoE3]D]Y_ZCjo.J\W&,JMVbe:l?1]H/H[iIMFOY@2*2 /StemV 109 "Y$l^/Q)?6;XG$'UB$`i6N'_95=bt'[l-D-*@=Z2) >> /Name/F12 22 0 obj 101 0 obj >> /Differences[49/one/two/three 65/A 114/r/s] `-d?R^seHO4$iNgY9^#U$/a.s&6#8m.;W*c-9-gM.tb#r$KfMkIrGA:BYo2]? 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0000080735 00000 n 102 0 obj Such problems are solved by matrix methods, so the material presented in this book is essential to these fields. /StemH 0 ,#2?qcjEsc_C;V"'4$gt>ta*MU@8s=FT3J>l`L%NRf+;\'7C.CV7ZaSM#@Jh66sb]B;ZYmV/.-D 83082 @s7(fV5nJBWZ":'J&K4'Lnm+XQIB`d[SG`*W178P3]D>:cMU!pZqF-6i 0000081902 00000 n /Type/FontDescriptor 0+LTcL?Lg]<8k$^YM$@kTKBK]#0K76_\pd]bN>YA(J]1AaGRu6!K/4p/ZIp]+$VZaQ20.Fh/Tn` << 1GTP`Cqsj!A+XK/IHF]%aJ?oW#9IRaAH~>endstream Introduction Formulating a project plan and completing the logical framework matrix requires some understanding of the logical framework approach to project planning. 82 0 obj endobj . b3dJpP39qT0WAV\,h$t41^iT@;FG.Kmjd=Vot;0c:249H/VrihrRjgI%sSmN%h(2,bRLmh`kLt= /Subtype/Type1 ]f;Kd-ZZJEbM#3bFS.MO7a'$QG5'fPPm:B1:%*r`6EOkZec`-sG+dr;L/Eb$]kuVc)\Baqg?.hP stream /FontName/JKZJDB+cmsy10 :-pR0Ds$eY/4F0h'X3k /LastChar 113 Ae+5.n"augRJK"[hN>.R:nY,q+9$?frn-'\?&XdJ,Jq7,g0@ddfX)YFb075jS'fH[SQNn,Yk0i] /Resources<> /Differences[34/quotedblright 40/parenleft/parenright 65/A/B/C/D/E 73/I 77/M 80/P 83/S 86/V 89/Y 92/quotedblleft 94/circumflex 97/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y 123/endash] /Ascent 683 ! endobj /CapHeight 683 /Length 539 fkgBD((7lbAc)* 5 0 obj The concept of matrices is so powerful, that in many cases, we make our lives simpler by viewing a vector as a special type of matrix. endstream 9rr(oRMU.FA3A-(S#=Z"WC64iV'_Y,Rcre;hU]O:1:Om9$jGZh3KoT.M(%<6Ohnl'bH\n4","AI /Subtype/Type1 )Oe$iS#d/hZrC 0000081902 00000 n /Name/F18 /StemV 47 /FontBBox[0 0 0 0] A matrix where every element is zero is called a zero matrix. 42 0 obj 0 117 48 0 obj /Ascent 683 $F)Vh%K)]IqFCQ=*Ge^Oqo+7. /ItalicAngle 0 [s*M1M?+o$m25;lo^#ok.WG7T_s>? 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/Encoding 91 0 R Matrices 4.1 Introduction to Matrices Objectives: • Students will organize data into matrices • Students will solve equations using matrices A matrix with only one row is called a row matrix. fl&T&l*I4c2M%;A 0000069764 00000 n /Ascent 683 0000051728 00000 n 9OL&pJoaLEQ-8;Jp8c_H3G>s)n+:A3?s_PgqN'.>*J6D\A%X[aoX.b),afn%P_fr!8')\)XAG3i /Flags 262178 rBFY3=C#I[-d:X&U)Z").JHR>;\U5r:1R>'$.MJ$6%#V0jJ^M:%BX@LV5=\2?DWO.k:@+/! /FontBBox[0 0 0 0] /Creator(dvips 5.72 Copyright 1997 Radical Eye Software (www.radicaleye.com)) 104 0 obj stream endobj 2 0 obj /Length1 875 endobj /FontFile3 60 0 R /Descent -194 0000041891 00000 n stream /BaseEncoding/MacRomanEncoding 0000007347 00000 n =G8&^I7R:F'ZT%Bbr[lQeBCcM^%\N4P!q,!t0f0JX0YW /Filter[/ASCII85Decode/FlateDecode] '=S;? 0000073053 00000 n )?~>endstream :$6A3eYa!JLm3FoPo :JsVj:S8F9oHZ@UO&]\-ChV`T4S!GN)E^,cE+D--G\KYHMkA*)3rEjfgT$g73mh+d:%8jTM=8TML >> /FontDescriptor 77 0 R "FC+`GNdSC'LoKV 52 0 obj >> LYMmP3ttJu.jD:T!$tC7W*PZRMmd:40tnk=J3P@gI*9O>0E$^ur=j#H/,#d&.0~> 0000041468 00000 n ;06(Z%[$HTe\-Yo[#gZp\H 0000053233 00000 n 3M'5BmV`I+XJDoPZl@i>?7NPe-V>$>l+;Xknucd=V;o^L[Y/. stream N#b8akl2cb_n0Kic"SO=lsp0Pl4dj&%6crZb3`PuB#a.Kd^AD=/md_elddga@cLou_M>Ps&7N^f /Type/Encoding /LastChar 112 &'b=QkSD(BN6h8)St\BHX'Va6a1E!&\IiQk["Se#KFN#,-YVu->,YNPrCgN);&Xab4j2=[>AEN6 0 117 /Flags 4 << ;Ai5T>sr"iD+W>>~>endstream << /BaseEncoding/MacRomanEncoding /Length1 875 /FontFile3 56 0 R VZZ+7EB5@uKlAN/aKGCb^5suL6JtN8`V72dE#=i?-ZJ6n^c3/Oo7^+YQa<<8mH9(R(4PJC~>endstream Introduction to Matrices 1. 5-/3eG,DbE3@Vp1VjWI7eAStis,9>]@j[rEd'+l>6&SBQHY[c2(.nih)Y]-sP_N(UnUJE e<4sVN/GMQD\9\"0-%_bg:.GW=[tq3,#*u1:2A"og0-sGb:W:JJtnpEn/F!_FlC]]RFe6i=nc>? << /Widths[434] >> Since the publication of the First Edition of this book in 1960, the field of matrix theory has expanded at a furious rate. 109 0 obj p3! endobj /Subtype/Type1C /Parent 3 0 R 0000078849 00000 n 0000025433 00000 n >> 66 0 obj mo,tI?`=m5j5l+uP[Vrc]q9%.Mo(A'[:0\1OfCU?QC"!\(D(.9s$UA3T3]+LZ*I_#b\"eI 0000045680 00000 n /Subtype/Type1C ZKUi,hu00KO5^isih?f&]*uBT:qT/md935N.snfMj@q@Oa+3. 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