# 1.1 - Dr Doom MATH185O Page 1 of 7 Chapter 1 Linear...

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Unformatted text preview: Dr. Doom MATH185O Page 1 of 7 Chapter 1 - Linear Equations and Matrices Section 1.1 - Introduction to Systems of Linear Equations Deﬁnition: A linear equation in 71 variables (or unknowns) 3:1, 3:2, . . . ,asn is an equation that can be expressed in the form: alga +a2\$2+...+ana§n = b, ‘F REAL nunbas Where a1,a2, . . . ,an, I) are constants (reals). The ai’s are referred to as co- efﬁcients. A non-linear equation is, surprise-surprise, one that cannot be written as such. Examples - linear / non-linear equations: 2:61 — 3362 + 5 = —11:C3 11w +140)“ 4— “\$3 = -—g LCNEAL ul.l..'\’ 7")“ ,x3 (ii) w — 33: + y — z = 0 U145». in w, “,1”, (m) 7193 — ﬂy + 2 = 103,2 Lula». m x W1 ( 1r, 41,631 MI as) (iv) 33y — 2,2 = 4 par LiMEAL m x ,vsn- ( = 5 Nﬂ Lm'cAz. uh xt ,1... U) ' ‘ 1. ‘5 (vi) a: — \$2 + 3\$3 = 12 Mar mire». m 76 (gut Lmem m x , x ,x) Q€AL Nov—Lb Maw-7M ' 11+ Wm M slings I HAUE \$400 To )ﬁuEsf u) \$TOCK A ,(5,C (\$1759 “’1 _ DEFQE x “an. As it ovum“ ot— «smcr. A‘gﬁ lesvectwew lOO= ZX+35451 w Tale cause we NEAL w'u-n u'ucAr. Faun-dons Dr. Doom MATH185O Page 2 of 7 Deﬁnition: A solution of a linear equation alga + @332 + . . . + anasn = b, is a sequence (or n—tuple) of real numbers 51, 52, . . . , 5” such that the following is true: alsl + a282 + . . . + ansn = b. The solution set (or general solution) of a linear equation is then the set of ALL solutions of that equation. Examples — ﬁnding a solution set: ﬁgs—334212 151' g—JL For. k some: m1. Number. ‘Tuueu sow: Fob x 1:) TEL/4,5 0F *5 éx-Bg—Al cs» 6x=m33<=° x: hi? So an :2, -\— i i (a: k, '1‘ “we swan scunioﬁ QAP—‘TH'LULAV- saLu-rio» «ssfeu 1: (2sz ii To* 9'! WE J’ch WT x= l’r‘i-‘HR. i=4- FOL A Ha.) 10 CW X=H, tad-t is A—saLu‘rioxl (if—34+: [L \/ Dr. Doom MATH185O Page 3 of 7 Deﬁnition: A system of linear equations (or linear system) in n vari- ables 331,332, . . . ,asn is a ﬁnite set of linear equations in those variables. A solution of a linear system (if it exists) is a solution common to all of the linear equations in the system. Examples - linear systems: a: — 2y + 3,2 = 2 33: + y = 4 y — 2,2 = —1 K=43‘3=4 ’24 Is A' sum-ion or. 'l’l-l-E 52mm- oue'oL » A“ ea'd l’ l-l+\$~\=1 ‘/ 2"” saw 34m =# / 3"” €Q‘IJ \‘1'\ =‘\ \/ Dr. Doom MATH185O Page 4 of 7 Deﬁnition: A system of equations that has no solution is said to be incon- sistent. A system of equations that has at least one solution is said to be consistent. Examples - consistent / inconsistent systems: . — = 10 .. — 2 = 1 -- 3 = 2 <2) 3” y ' (u) 3” 9 © (m) 3:” y a: + y = [email protected] —23: + 4y 2 3(9 63: —— 6y 2 4 CN) Abb ®+© . Q3) l®+® 6N6!) zP°eQuA1CoA 16 12.6w»me 2.x 4! 0% = LI 0 O S ) 7” 3° 7‘: (9:61 —:. L‘éT ‘9 5a )6 l— 0». 0" ‘3 'We” Vi : x4“ wan-cu k Aasuw 1%” 7‘: —3+3§ sysren is cal-’5‘. _ 5° Sys-rE/‘t is (Limusa's‘revr ‘6 '6 a3 LUBES A027 // up Coan “NETS. Dr. Doom MATH185O Page 5 of 7 Theorem: A system of linear equations has either no solution, one solution, or inﬁnitely many solutions. Examples: . a: — 2y 2 3 x—lm’js (Z) —333 + 6y 2 —9 or) 0:0 CAN) 311N123 1&6 4‘ EmeéA 101m: swam: GNN) L117 xz‘k C£é(2) was at: 13+} 3.1%»(5 576nm ems co'A/IAUT sat—URQAS. /_ .. a: —— 2 = 5 <2» 2m __ 2 3 A» e): 1.. 2W Own-0.3 r— —'7 ol— Oa—‘l waiw is “same. no soLuﬁod _/ \$1 + \$2 —— \$3 = 0 - _ (iii) :01 _ 333 = 0 X\=)€-L=x5=O ls A sowrmﬁ ‘ \$2 —— x3 = 0 ii- is we omQuE scum» (wc'LL see war we.) Dr. Doom MATH185O Page 6 of 7 Augmented Matrices Deﬁnition: Given an arbitrary linear system of m equations in n unknowns: a11\$1 -- (“2:62 -- . . . -- @1713?” 2 b1 (@1361 -- (@2362 -- . . . -- @2713?” 2 b2 amlgcl __ am2\$2 __ - - - __ amngcn = bm we have the associated augmented matrix: all Q12 . . . am b1 am am . . . CLQn b2 aml am2 - - - amn bm Examples: (i) From system to augmented matrix: 233 — y —— 2,2 = 10 z .4 1. \0 a: + 3y —— z = —2 —> \ 3 \ '2— 33: —— 3,2 = 7 3 o z, 7 (ii) From augmented matrix back to system: 2 0 1 0 3 3 be +>‘3 +3>c< =3 1 1 3 2 0 0 —> ¥l+x1+5xﬁ2¥q = O 4 —1 —1 0 1 —2 fo|~¥1"xs + x5 "7- Dr. Doom MATH1850 Page 7 of 7 Operations on equations: 1 Multiply an equation thru by a non-0 constant 2 Interchange two equations 3 Add a multiple of one equation to another Example 23: — y —— 2,2 = 1 33 + y —— z = —1 2y — 3,2 = 1 - sun'Tu—t cam-n5»: I. a z - AN) 62);“ ear.) To me? 2"“ mm. X-kuaa—L =4 .33 = 3 23’32= 4 ' so \ -:.L. o Muunvw 2 662A 13y 3 )4 + :3 +2. —.-_ -—l 3 = " .2\a’32 =' 4 50L‘ 5) 2: ~32: 3 MvL-néu/ ‘2,” 5a") (57' ~11}- ¥+ +2--\ :1 'z. a '4 Is x5| gc'l 1:4 Elementary Row operationiz‘ on MATMUS 1 Multiply a row by a non-0 constant 2 Interchange two rows 3 Add a multiple of one row to another. o I o -) o o 1 " we‘u. be A LOT X: 4 no“ op Thus ‘3’" may: mm z=’l ...
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1.1 - Dr Doom MATH185O Page 1 of 7 Chapter 1 Linear...

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