Lhaptel This fast You're will I into be review a - for - PowerPoint PPT Presentation

- Fly through Lhaptel This fast You're will I into be review a - for reading Chapter 2) ( and responsible 1 and if something talking to there's me you don't follow . : big ideas / terms " set ( Think like " the not , , , proof of Proposition . ) intricacies of 1.30 say . . , * of solving systems I'll post review a of swbst 'n linear eqns using elimination , multiplication matrix or .

Fundamentals / Very f for for all : every , ( F ! antique ) F : there there exists exisb : iff and only if iff ( ⇒ : , We much depth won't sets in Mostly : use . : XHR } numbers fx R = ieal 1122¥ x. YER } ( x. g) : of those subsets and esp .

RecaH"Abstat"FunctionNotal= f B domain A A → : : ( image , xtsftx domain range ) B ) co : fan target ( space ) × ~ T " to " maps f > IR : R xzo } E± { XEIR - or : > ×2 X ~ injective ( l :L ) Det two one-to-one domain elb : in or sent to different outputs . swjective onto : of co domain ett is hit or every we .

Injective Notate Sejdiu Functions : . 3 . ( bisection ) 4 . 5 . fal=u 6 ,HtH,H=fti ) . 7 . 8. : 0 .

Vectors Points and Lines , ÷ pair of real # a , b) 2D) ordered Vector is ( ' an s , . =) ( a ,b ) ( a , b) Common notations : , U ( ui Our book ua ) = : µ , interpretation , Virtually in Graphically U is an arrow : - - - - . . , . ui ( here 0 ) , Ua ) Ui algebraic defy has vector concept every an geometric / axiomatic and one a . Vi int

odg geo → ¥4 :b Addition Utkfuuytuuattthfu ( scalar ) ¥ cU=du , ,ud mwlth , ,cuI =L cu Eu Subtraction V=UtHW U . v , ,Usy) =L u , - # U=cVwV=cU linearly ( parallel ) Ia ,b dependent that such 01=10,0 ) ) aUtbV= a ,b both not 0 .

odg gee U .V=lu dot product , ,uD•(v , ,v , ) : ( inner product = + Uava Hiv , , scalar product ) ( u ,v , I Usb ) , . UnV=( u ,v ) length Hull .in#g th # " magnitude

€ If you learned from Stewart 's book vectors . . . make huge distinctions between points and Stewart vectors : /¥f n = ( 3,2 ) • P # = OF ( 3 , 2) =) and point We won't For Vector us are synonyms . , . - It's clear from and it context makes , life to do it this To easier wit way : .

Det the Given a point P and set vector U non zero . , se R } l={P+sU : - line is U a Post 's U•• optrou • • P•t2U P P 2U Ptu - ( dir l direction indicator vector ) Points A is a on are " incident . " with line . incident w/ all of them if F collinear , Pn R line , Ps are , ... . line ( , 2) + sl 3 ( k , 0 ) s= I ) - 4) H EI is on , 1+35=5 ys=6 } sotn - ( 5,6 ) not no is 2 . .

Pts ( Q ' P ) t.fi p=U a- µ than • FQ=QI . µ P ¥ " • } , } { Qtslpal gp+s( / 5.0 = 'QF so ' ' pot ' ' ' PI # OF Hxlkfax length of PI . pH=HP HQ . QH is =(Q-P,Q# FIAT =

we expect ? Does this def " everything cover Can get segments , values restricted of using • rays s . Yes ( wksheett farm Two points line ? • a PIOPII two DI 's of line vectors zero are non same - if± they 're Mutt 's of each other scalar . Then F unique line # Let incident Pt Q . ' POT both of and with U = Q - P DI is a , , ' difference of two pts of POT DI every is line the on . P %% • slope ¥ if " of 2 Pts U is , to u : • , . U ,

other forms : eliminate parameter - 4) ( -1,2 )+s( 3 ( 4s ) -1+3 2- E± - , , IF It 5=51×+1 ) X=3s -1 ⇒ - fly - 2) -45+2 ⇒ y= s = - t.ly - 2) f- ( x + D= 4×+4=-3+6 solve for y . . . 3y= -4×+2 - 31×+3 y=

, l//m if Det Two lines l are parallel H DI 's are m , . l= { Pts U } m= { QTTV } Pap Lines # , 11 ) A one point if , V tin indep ( not U • in . . empty UHQP UHV and if Ah y , p ¥ • OF u||✓ if line • are same and UHQ - P K ;←Q± ;± .

Warm-up Pkn ( 4131171 . ( v , , b) UaV= ( u , ,u2 ) Red Uzva U , v. + = : Proves : The dot product U V= V. U commutative is . : ' V V. U U U ,v V , U , Valls uzvz + = + = = , . ( Vtw )=U.V+U distributive Proves : The dot product . W U is : . ( ✓ t W ) • ( v , u Ua ) ( U Vatwa ) = + w , , , , x. ( v. , ) ualvatwa ) w - = + + Uaw wit U = + U2V2 U , V , 2 , = ( U t ( tuzws ) uava ) , v , + u , W , . V . W U U + =

Quickstatuscheck which of Axioms Euclid 's : far ? work so ✓ line containing them F Given two pts , � 1 � ✓ extended indefinitely � 2 � Lines be can ✓ � 3 � Given - All A ,B , F circle cent 'd at HD A r= - AH=r } C={ X AI : HX radius with A. . • B - � 4 � Right angles X are all equal Hw ) ( ✓ || postulate � 5 �

lines l Two U , v + uava = = o are m , . , perpendicular if they Indicators 1- Direction have . Pe_repend@ty1o_thogma_ityDefU1VifU.V |/ and -1 play important roles . . . Wry point , 7 exactly # If l line and P is is a one a line incident P and H l with to . point , 7 exactly If l line and Prep # P is is a one a line incident P and with l 1- to .

point , F exactly If l line and Pap # P is is one a a line incident P and l with -1 to . •¥# + SU l Pf Suppose Ufo Q : , V= ( , .ua - us ) , u.lt U Claim U=lu ⇒ : veutw I U l show is any now o ) V= tua Cased U= ( u mull , .ua ) scalar ⇒ u ,=o . , , of this . UT . V= ( o uol.L.ua , D= ✓ U 0+0=0 ✓ , → (ase2_ Uz=O . tus . ✓ @ gen 'l (ase3_ , 4) U , UD Aiustu in = - ,ua=o = , , + SV P l incident wlp 1 m : , .

Given lined NormalFoI choose Yet All and , \f%× ( i.e. ofl ) Then ALU DI U any . , g a Aceto } th : of line nleqn ( • + L x. yi . ,xa ) ( ) Remember A fixed ; , Y X= ( x = " special is variable this HAH =L " nlegn If Is . , . . ( lx • ( X - 13,1 ) ) Y=( 3. D , - 1,2 ) - y ) =L EI A =L ( -1,21 - I. 2) , ,xn . ( = ( -1,21 - l ) x , -3 Xa ¥t¥ , 2=0 - × , +3 +2×2 = - ⇒ III. tartan 's ,

F alternate of normal farm l version : A \•f 't yko A. lx . t - AaY=O A. X A. ( x ,y ) =0 C - [ A- ( x . ,xd=c] An X=c .x+i=o A=H,2) 4=13,1 ) EI , - y )=o A. 1 x A X Ito ⇒ A - A .X= - 1 -1 =

Between Let fcs ) fun fcsI=Q Then be egn of line , =P . , is between f- C Ss ) R if P < 52 , R F , Q sz , s s ] IEHEOP , ( = . the Q : fcsas • fo , p . . Given ' Corollary 1.21 a line 3 pts on , blw other two • must be One . • • < for further See book corollaries with 4+ points

" half planes A line separates " 1122 into two . 7 L , Qctl opposite sides of l Clever Def Rtl P if F on - them between Otherwise , the they 're same on . side . l p l •\•Q R P¥ • ¥0 l , .

Prep Qttl l - YKO Atl ) and A. l x l Ytl then # P let : , , . if same / opposite side of l P and Q are on Hafts A • ( P - Y ) , A. ( Q - Y ) have signs same . D. . X A Book AP A. Q C = uses compares c c - - , , . D. " simple , simpler ) Would be if had " 1 well we = HEH . HBH for E. I cos A Oeco , Mad > 0 it ] Otltla for < 0 , - . P • l Y

- YKO Atl ) and Prep l A. ( x ( Ytl Qtl Then # P let : , , . side of if P and Q same / opposite l are on - Y ) Aa ( P - Y ) Aol Q have 1 opposite sign same , . = A. ( Cpi Y if # ) ) ] for PI Let oesel go ) - PI line segment for iff glsko PI l at pt R intersects gcst.to#ED+sA.l+Q-PT=b+ms=ms+b some s ( sides ) P , Q opposite ⇒ Now That 's linear values Af minlmax so P • , @ endpts gcsko only , Y have diff glol , ga ) . signs a • . - YI A. ( Q A. ( P - y ) g (1) glo ) = = , .

Second Warmup Question : . ( CU ) .V= c ( U .V ) Prove : = ( - ✓ EU ) , ,c uol.lv PI , ) , ,v ( u : C = ( UN t U2V2 , C ( uzva ) = au t , c ( uv ) =

Moving # kapter2= You read § 2.1 " Other Matrix than Concepts on . - need multiplication for matrix we 'll all now : . U ,VeR2 be linearly independent Then Lemmond Let . V. XE R3 F be IR that such unique a , add " " X cards ( a. b) = combination linear of U , V . . ) ( See formulas for . in particular lemma for a ,b . .

in IN ULV Let XEIR ? Then Lemma2= Let , Vto U with , . x= Yiutrutiftfzv Corollary -2.5 conditions ) 11×11 ? l×juh÷ + Ynys ( same

Distance qualities " 2 = ( x. x 7 Hxlkxfx We defined Also , = IPII - QH = / OF / HQ - PH dist ton H P P Q to = = V¥kltu a RJ IPFI = IRJI Def FQ if qngment_ are . Det Congruence of line segments is equivalence relation .

Quick#d Equivalence Relations : Examples of Relations 2 R if 541 aRb a2=b2 3<4 < : , , , 3=3 3=14 IR , : = , if A relh equivalence rekn it is is an ... : tfx × Rx reflexive � 1 � , � 2 � symmetric × Ry if then yRx V × , y , : . if xRy Vx and yRz , then transitive xRz � 3 � : ,y,z ,