UNIT FIVE DETERMINANTS


 Sharleen Barnett
 1 years ago
 Views:
Transcription
1 UNIT FIVE DETERMINANTS. INTRODUTION I uit oe the determit of mtrix ws itroduced d used i the evlutio of cross product. I this chpter we exted the defiitio of determit to y size squre mtrix. The determit hs vriety of pplictios. The vlue of the determit of squre mtrix A c e used to determie whether A is ivertile or oivertile. A explicit formul for A exists tht ivolves the determit of A. Some systems of lier equtios hve solutios tht c e expressed i terms of determits.. DEFINITION OF THE DETERMINANT Recll tht i chpter oe the determit of the mtrix A ws defied to e the umer d tht the ottio det A) or A ws used to represet the determit of A. For y give mtrix A [ ij ], the ottio A ij will e used to deote the ) ) sumtrix otied from A y deletig the i th row d the j th colum of A. The determit of y size squre mtrix A [ ij ] is defied recursively s follows. Defiitio of the Determit Let [ ij ] A e mtrix. ) If, tht is A [ ], the we defie det A). ) If >, we defie deta) ) deta ) Exmple If A [], the y prt ) of the defiitio of the determit, det A). If A, the y prts ) d ), det A) ) )det[] ) )det[] ))) ))) If A, the usig prts ) d ), we clculte the det A) s follows.
2 )det ) )det ) )det ) deta) )) ) )) ) )) ). ofctor If A is squre mtrix, the ij th cofctor of A is defied to e ) ij deta ij ). The ottio ij will sometimes e used to deote the ij th cofctor of A. Exmple Let A. The ) det ) ), ) det ) ) d ) det ) ). I the defiitio of the determit, prt ) cosists of multiplyig ech first row etry of A y its cofctor d the summig these products. For this reso it is clled first row cofctor expsio. Exmple Let A. Use first row cofctor expsio to evlute deta). Solutio deta) )det ) )det ) )det ) )det ) )) )det ) )det ) )det ) )) )det ) )det ) )det ) )) )det ) )det ) )det ) )) )det ) )det ) )det ) )){)) ) )) ) )) )} )){)) ) )) ) )) )} )){)) ) )) ) )) )} )){)) ) )) ) )) )} )){ } )){ } )){ } )){ }
3 Although the defiitio of the determit uses first row cofctor expsio, the determit of A my e clculted y tig y row or colum) d multiplyig the etries of tht row or colum) y their cofctors d summig the products. This result is give i the ext theorem whose proof is omitted. Theorem Let A e squre mtrix, the i j deta) ) idetai ) ) jdeta j ). i th row cofctor expsio j th colum cofctor expsio Exmple Let A. Evlute det A) y ) secod row cofctor expsio. ) third colum cofctor expsio. Solutio ) det A) ) )det ) )det ) )det )) ) )) ) )) ). ) det A) ) )det ) )det ) )det ))) )) ) )) ). Theorem If A is squre mtrix cotiig row or colum) of zeros, the deta). Proof Use cofctor expsio log the row or colum) of zeros. Theorem If A is mtrix with two ideticl rows or colums), the det A). Proof The theorem is certily true for sice det. If, use cofctor expsio log the row differet from the two ideticl rows. Let this row e the th row. Usig cofctor expsio log this row gives det A) ) )deta ) ) )deta ) ) )deta ). But ech of the sumtrices A, A d A hs two ideticl rows so their determits re, hece det A) for y mtrix. If > proceed s ove writig det A) s sum of products ivolvig sumtrices with two ideticl rows whose determits re.
4 Trigulr d Digol Mtrices A squre mtrix is sid to e upper trigulr mtrix if ll the etries elow the mi digol re zero. A squre mtrix is sid to e lower trigulr mtrix if ll the etries ove the mi digol re zero. A squre mtrix is sid to e digol mtrix if ll etries ot o the mi digol re zero. A digol mtrix is oth upper trigulr d lower trigulr. Exmple is upper trigulr mtrix. is lower trigulr mtrix. is digol mtrix. It is oth upper d lower trigulr. Theorem If A is upper trigulr, lower trigulr or digol, deta). Proof Suppose A is upper trigulr. To evlute deta) use cofctor expsio log the first colum. Sice there is oly oe ozero etry i the first colum the expsio gives deta) ) deta ) deta ). Now A is upper trigulr so proceed s ove to use cofctor expsio log its first colum to get det A ) deta ) where A is A with its first row d first colum deleted. omiig the results gives deta) deta ). otiuig i this fshio, we evetully get det A) s required. If A is lower trigulr or digol, the rgumet is similr. Exmple det ))). det ))) Theorem deti ) for ll. Proof Sice I is digol mtrix, deti ) )) ) BsetWeve Method The followig method is ltertive wy to evlute the determit of mtrix. This method is oly pplicle to mtrices d is sometimes clled the setweve method. ostruct rry y writig dow the etries of the mtrix d the repetig the first two colums. lculte the products log the six digol lies show i the digrm. The determit is equl to the sum of products log digols leled, d mius the sum of the products log the digols leled, d.
5 Exmple Use the setweve method to clculte the determit of A. Solutio det A) ) ) ) ). PROBLEMS. Use the defiitio of the determit to evlute the determit of the give mtrix. ) ) c) d) e) f) g) h). Use the setweve method to evlute the determits ), ), c) d d).. Evlute the determits of the followig mtrices y ispectio. ) ) c) d). Let. A Fid the followig cofctors of A. ) ) c) d) e) f). Fid ll vlues of for which  det.
6 . ELEMENTARY ROW OPERATIONS ON DETERMINANTS The evlutio of the determit of mtrix usig the defiitio ivolves the summtio of! terms, ech term eig product of fctors. As icreses, this computtio ecomes too cumersome d so other techique hs ee devised to evlute the determit. This techique uses the elemetry row opertios to reduce the mtrix to trigulr form. The effect of ech elemetry row opertio o the vlue of the determit is te ito ccout d the the determit of the trigulr mtrix is evluted y fidig the product of the etries o the mi digol. Theorem If A d B re squre mtrices d B is otied from A y iterchgig two rows or colums) of A, the det B) det A). Proof Let A d B e mtrices. If, the A d B so detb) ) deta). If, the we use cofctor expsio for B log the row tht ws ot iterchged. Let this e row. The detb) ) detb ) ) detb ) ) detb ). Ech sumtrix B j is the sumtrix A j with its rows iterchged so detb j ) deta j ). Hece detb) ) )deta ) ) )deta ) ) )deta ) ) [ ) deta ) ) deta ) ) deta )] ) deta) det A) s required. If > proceed s ove usig cofctor expsios log rows tht were ot iterchged to get the fil result. The proof for iterchged colums is similr. Theorem Let A d B e mtrices with B otied from A y multiplyig ll the etries of some row or colum) of A y sclr. The det B) det A). Proof Suppose B is otied from A y multiplyig the etries of the j th row of A y. Use cofctor expsio for B log its j th row to evlute the determit of B d otig tht these cofctors for B re equl to the correspodig cofctors for A we get det B) ) j j detb j ) ) j j detb j ) ) j j detb j ) ) j j deta j ) ) j j deta j ) ) j j deta j ) [ ) j j deta j ) ) j j deta j ) ) j j deta j )] deta). The proof i the cse where B is otied from A y multiplyig colum of A y is similr. is commo fctor i row of the give mtrix. Exmple det det ) ) orollry det A) det A). Proof Sice ll rows of A re multiplied y the sclr to get A, usig the ove theorem times gives det A) ) ) ) det A) det A).
7 is fctor i row d row Exmple det )) det det ) ) )) Exmple Let A e mtrix d let deta). Fid deta). Solutio deta) deta) )) sice mtrix hs ideticl rows Exmple det det )). Notice tht row of the origil mtrix is times row. This leds to the followig corollry. orollry If A is squre mtrix tht hs row or colum) tht is sclr multiple of other row or colum), the det A). Proof Suppose the j th row of A is times the i th row of A. The det A) det Â ) where Â is the mtrix A with the j th row multiplied y /. But Â hs two ideticl rows, row i row j), so det Â ). Hece det A) det Â ). Theorem Let A e squre mtrix d let B e the mtrix otied from A y ddig multiple of oe row or colum) of A to other row or colum) of A. The det B) det A). Proof Suppose B is otied from A y ddig c times row i to row j. Evlute det B) usig cofctor expsio log row j. The detb) j j ) jdetb j ) ) c i j )detb j ) c i j )deta j ) sice B j A j for ll. j j c ) i deta j ) ) j deta c det Â ) det A) where Â is otied from A y replcig the j th row of A y its i th row. c det A) det Â ) ecuse Â hs two ideticl rows det A) j ) j )
8 Exmple Evlute det Solutio det ))det ))det )det ))))). The previous exmple outlies efficiet techique usig elemetry row trsformtios to evlute the determit of squre mtrix. The procedure cosists of usig elemetry row trsformtios to trsform the give mtrix ito trigulr mtrix i the ove exmple ito upper trigulr mtrix), tig ito ccout the effect of ech trsformtio, the filly evlutig the determit of the resultig trigulr mtrix y multiplyig the etries log the mi digol.. PROBLEMS. Use elemetry row opertios to evlute the determits of the followig mtrices. ) ) c) d) e) f) g). PROPERTIES OF DETERMINANTS Let A e squre mtrix. Let Â e the mtrix resultig from performig oe or more elemetry row opertio o A. Sice the effect of performig elemetry row opertio o the vlue of the determit is either to reverse the sig or multiply the vlue of the determit y ozero umer, d sice the elemetry row opertios rows & iterchged row hs fctor multiples of row dded to rows & product of etries o mi digol
9 re ivertile opertios; therefore det Â) if d oly if deta) d similrly det Â) if d oly if deta). Let sequece of elemetry row opertios e performed o the mtrix A so s to reduce A to its reduced rowechelo form R. Now A is ivertile if d oly if R I. But det R) det I) if d oly if det A). We therefore coclude tht A is ivertile if d oly if deta) d stte this result i the form of theorem. Theorem The squre mtrix A is ivertile if d oly if det A). Exmple det so is ivertile mtrix. det so is ot ivertile. A direct cosequece of the ove theorem is the followig result. Theorem Let A e squre mtrix. The the lier system Ax hs uique solutio for every if d oly if det A). Proof Suppose det A), the A is ivertile. The A is solutio to Ax sice AA ). To show tht this is the oly solutio to Ax, suppose tht x is lso solutio to Ax. The Ax so A A x ) A d hece x A. This shows tht A is the uique solutio to Ax. O the other hd, if Ax hs uique solutio, the whe solvig this system y mtrix methods the coefficiet mtrix is reduced to the idetity mtrix I d so A is ivertile d hece deta). Theorem Let A e mtrix d let E e elemetry mtrix. The detea) dete) deta). Proof The proof cosists i showig tht the result is true for ech oe of the three types of elemetry mtrices. Let E e the elemetry mtrix otied from I y iterchgig two rows of I. The EA is the mtrix resultig from iterchgig the correspodig two rows of A. The detea) deta) ) deta) dete) deta) sice dete). Let E e the elemetry mtrix otied from I y multiplyig the etries of some row of I y ozero sclr. The EA is the mtrix resultig from multiplyig the etries of row of A y. The detea) deta) dete) deta) sice dete). Let E e the elemetry mtrix otied from I y ddig multiple of oe row of I to other row of I. The EA is the result of ddig multiple of row of A to other row of A. The detea) deta) ) deta) dete) deta) sice dete). The result i the ove theorem c e geerlized to y two is omitted ut stted i the followig theorem. mtrices. The proof
10 Theorem If A d B re squre mtrices of the sme size, the detab) deta) detb). Exmple Let A, d let B, the AB. deta) detb) d det AB) deta) detb) )) detab). Theorem If A is ivertile mtrix, the deta ) /deta) Proof A  A I deta  A) deti) deta  )deta) deta  ) / deta). Theorem If A is squre mtrix, the deta T ) deta). Proof A cofctor expsio log the first row of A T gives the sme terms s cofctor expsio log the first colum of A. Exmple Let A, the A T. deta) )) )). det A T ) )) )). So det A) deta T ). Theorem Let A d B e squre mtrices with AB I. The BA I. Proof We first show tht there exists mtrix such tht A I d the show tht i fct B. Sice AB I d sice deti), therefore deta). But deta T ) deta), so deta T ) d hece A T is ivertile. Let D deote the iverse of A T ; so D A T ). T T T T T T T T T The A D I A D) I D A ) I D A I so D T. We ow show B s follows. I AB) A)B IB B. Whe computig the iverse of mtrix A oe should verify the correctess of the computtio y demostrtig tht oth the products AA d A A equl I. The precedig theorem proves tht i fct tht it is sufficiet to show tht oly oe of these two products eeds to e show equl to I.
11 . PROBLEMS. Determie whether the mtrix is ivertile or ot y clcultig its determit. ) ) c) d) e). Use determits to show tht the followig systems of lier equtios hve uique solutios. x y ) ) x y z x y z x y c) x y z d) x y z x y x y x y z x y z. Let A d B e mtrices with deta) d detb). Fid the followig. ) detab) ) deta ) c) detab ) d) detab) e) deta T B). Let A d B e mtrices with deta) d detb). Fid the followig. ) detab) ) deta ) c) detab ) d) detab) e) deta T B). THE ADJOINT MATRIX Recll the th row cofctor expsio of mtrix A for ws defied to e deta) ) deta ) ) deta ) ) deta ) where j the qutity ) deta j ) is clled the j th cofctor of A. To simplify our ottio we will deote this qutity y the symol j. Thus the th row cofctor expsio for deta) c e writte more simply s deta). Suppose tht i this expressio we replce the th row etries,, y the j th row etries j, j, j to get j j j. Such expressio would rise if the etries of the th row of A were replced y the etries of the j th row of A d cofctor expsio log this ew th row were doe. But the vlue of this determit would e sice the mtrix hs two ideticl rows rows d j re sme). We hve thus estlished the followig theorem. Theorem If A is squre mtrix, the deta) if i i j if i j j I similr fshio, we c deduce the followig result for colum cofctor expsio.
12 If cofctor expsio log colum is used the deta) if i j i j. if i j ofctor Mtrix Let A e squre mtrix. The cofctor mtrix of A, deoted cofa) is the mtrix otied from A y replcig every etry of A y its cofctor. Exmple If A, the cofa). Adjoit Mtrix If A is squre mtrix, the djoit of A, deoted dja) is the trspose of the cofctor mtrix; tht is dja) [cofa)] T. Exmple otiuig with the previous exmple, A, cofa) d so dja) [cofa)] T osider ow the product A dja) But deta) )det )det )det )) )) )). We see tht the product This exmple suggests the followig theorem. A dja) is digol mtrix with the digol etries deta).
13 Theorem If A is squre mtrix, the I deta) dja) A Proof dja) A I deta) deta) deta) deta) deta) deta) deta) Theorem If deta), the A is ivertile d A dja). deta) Proof From the previous theorem we hve I deta) dja) A. Sice deta) we c divide y deta) to get I dja) deta) A so A dja). deta) Exmple Use the precedig theorem to fid A for the mtrix. A Solutio This is the sme mtrix used i the previous exmple where we foud dja) d deta). Usig A dja) deta) we get A / / / / / / / / /.
14 . PROBLEMS. For ech of the followig mtrices fid its djoit, the use the djoit d the vlue of the determit) to fid the iverse of the mtrix. ) ) c) d) e) f). Let A e mtrix with deta). Show tht det[dja)] [deta)].. Let A e mtrix with deta). Fid det[dja)].. Let A e mtrix with deta). Fid det[dja)].. Let A e mtrix with deta). Fid det[dja)].. RAMER'S RULE rmer's rule provides formul for solvig system of lier equtios i vriles whe the system hs uique solutio. Theorem rmer's Rule) Let Ax e system of lier equtios i vriles with deta). Let A e the mtrix otied from A y replcig the th colum of A y the colum vector. The the system hs the uique solutio deta x ),,,,. deta) Proof Sice deta), A is ivertile d the system Ax hs the uique solutio x A. Therefore x dja) deta) deta)
15 deta). Thus.,,, for deta) x Now the umertor of x cosists of the th colum cofctors of A multiplied y the correspodig etries of. We get the sme result if we use th colum cofctor expsio of A so deta) ) deta x for,., Exmple Use rmer's rule to solve the followig system of lier equtios. z y x z y x z y x Solutio det det x det det y det det z. PROBLEMS. Use rmer s rule to solve the followig systems of lier equtios. ) x y ) x y c) x y x y x y x y d) x y z e) x y z f) x y z x y z x y z x y z x y z x y x y z
16 . APPLIATIONS USING THE DETERMINANT I this sectio two pplictios usig the determit will e cosidered. The first is coveiet method for evlutig cross product. The secod pplictio is fidig the volume of prllelepiped. ross Product Recll if u u, u, u) d v v, v, v ), the u v uv uv, uv uv, uv uv ) uv uv ) i uv uv ) j uv uv ) where i,, ), j,, ) d,, ). A coveiet method for rrivig t the cross product u v is to use the mtrix form i j u u u d do first row cofctor expsio s you would do to evlute v v v determit. Exmple Let u,, ) d let v,, ). Fid u v. Solutio i j u v i j i j,,) Volume of Prllelepiped Recll tht if prllelepiped hs the vectors u, v d w s edges, the the volume of the prllelepiped is w u v). A stright forwrd w w w u u u clcultio shows tht w u v) det u u u det v v v v v v w w w Hece the volume of the prllelepiped c e foud y tig the solute vlue of either of the ove determits. Exmple Fid the volume of the prllelepiped hvig the vectors,, ),,, ) d,, ) s edges. Solutio Volume det.
17 . PROBLEMS. lculte the followig cross products. ),,),, ) ),,),,) c),,),, ) d),,),, ). Fid the volume of the prllelepiped hvig the followig vectors s edges. ),, ),,, ),,, ) ),, ),,, ),,, ) c),, ),,, ),,, ) d),, ),,, ),,, )
Chapter 04.05 System of Equations
hpter 04.05 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vicevers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More informationGaussian Elimination Autar Kaw
Gussi Elimitio Autr Kw After redig this chpter, you should be ble to:. solve set of simulteous lier equtios usig Nïve Guss elimitio,. ler the pitflls of the Nïve Guss elimitio method,. uderstd the effect
More informationArithmetic Sequences
Arithmetic equeces A simple wy to geerte sequece is to strt with umber, d dd to it fixed costt d, over d over gi. This type of sequece is clled rithmetic sequece. Defiitio: A rithmetic sequece is sequece
More informationRepeated multiplication is represented using exponential notation, for example:
Appedix A: The Lws of Expoets Expoets re shorthd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you
More informationA function f whose domain is the set of positive integers is called a sequence. The values
EQUENCE: A fuctio f whose domi is the set of positive itegers is clled sequece The vlues f ( ), f (), f (),, f (), re clled the terms of the sequece; f() is the first term, f() is the secod term, f() is
More informationChapter Gaussian Elimination
Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio
More informationA black line master of Example 3 You Try is on provided on page 10 for duplication or use with a projection system.
Grde Level/Course: Algebr Lesso/Uit Pl Nme: Geometric Sequeces Rtiole/Lesso Abstrct: Wht mkes sequece geometric? This chrcteristic is ddressed i the defiitio of geometric sequece d will help derive the
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2πperiodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL  INDICES, LOGARITHMS AND FUNCTION This is the oe of series of bsic tutorils i mthemtics imed t begiers or yoe wtig to refresh themselves o fudmetls.
More informationSTUDENT S COMPANIONS IN BASIC MATH: THE SECOND. Basic Identities in Algebra. Let us start with a basic identity in algebra:
STUDENT S COMPANIONS IN BASIC MATH: THE SECOND Bsic Idetities i Algebr Let us strt with bsic idetity i lgebr: 2 b 2 ( b( + b. (1 Ideed, multiplyig out the right hd side, we get 2 +b b b 2. Removig the
More informationSummation Notation The sum of the first n terms of a sequence is represented by the summation notation i the index of summation
Lesso 0.: Sequeces d Summtio Nottio Def. of Sequece A ifiite sequece is fuctio whose domi is the set of positive rel itegers (turl umers). The fuctio vlues or terms of the sequece re represeted y, 2, 3,...,....
More informationLinear Algebra II. 4 Determinants. Notes 4 1st November Definition of determinant
MTH6140 Liear Algebra II Notes 4 1st November 2010 4 Determiats The determiat is a fuctio defied o square matrices; its value is a scalar. It has some very importat properties: perhaps most importat is
More informationEXPONENTS AND RADICALS
Expoets d Rdicls MODULE  EXPONENTS AND RADICALS We hve lert bout ultiplictio of two or ore rel ubers i the erlier lesso. You c very esily write the followig, d Thik of the situtio whe is to be ultiplied
More informationSect Simplifying Radical Expressions. We can use our properties of exponents to establish two properties of radicals:
70 Sect 11.  Simplifyig Rdicl Epressios Cocept #1 Multiplictio d Divisio Properties of Rdicls We c use our properties of epoets to estlish two properties of rdicls: () 1/ 1/ 1/ & ( ) 1/ 1/ 1/ Multiplictio
More informationGeometric Sequences. Definition: A geometric sequence is a sequence of the form
Geometic equeces Aothe simple wy of geetig sequece is to stt with umbe d epetedly multiply it by fixed ozeo costt. This type of sequece is clled geometic sequece. Defiitio: A geometic sequece is sequece
More informationDEFINITION OF INVERSE MATRIX
Lecture. Iverse matrix. To be read to the music of Back To You by Brya dams DEFINITION OF INVERSE TRIX Defiitio. Let is a square matrix. Some matrix B if it exists) is said to be iverse to if B B I where
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More information1 The Binomial Theorem: Another Approach
The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets
More informationShowing Recursive Sequences Converge
Showig Recursive Sequeces Coverge Itroductio My studets hve sked me bout how to prove tht recursively defied sequece coverges. Hopefully, fter redig these otes, you will be ble to tckle y such problem.
More informationSection 3.3: Geometric Sequences and Series
ectio 3.3: Geometic equeces d eies Geometic equeces Let s stt out with defiitio: geometic sequece: sequece i which the ext tem is foud by multiplyig the pevious tem by costt (the commo tio ) Hee e some
More informationDETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.
Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of
More informationMath Bowl 2009 Written Test Solutions. 2 8i
Mth owl 009 Writte Test Solutios i? i i i i i ( i)( i ( i )( i ) ) 8i i i (i ) 9i 8 9i 9 i How my pirs of turl umers ( m, ) stisfy the equtio? m We hve to hve m d d, the m ; d, the 0 m m Tryig these umers,
More information8.3 POLAR FORM AND DEMOIVRE S THEOREM
SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,
More informationSection IV.5: Recurrence Relations from Algorithms
Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by
More informationMATH 90 CHAPTER 5 Name:.
MATH 90 CHAPTER 5 Nme:. 5.1 Multiplictio of Expoets Need To Kow Recll expoets The ide of expoet properties Apply expoet properties Expoets Expoets me repeted multiplictio. 3 4 3 4 4 ( ) Expoet Properties
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationBINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients
652 (1226) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationDeterminants. Dr. Doreen De Leon Math 152, Fall 2015
Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationSolving DivideandConquer Recurrences
Solvig DivideadCoquer Recurreces Victor Adamchik A divideadcoquer algorithm cosists of three steps: dividig a problem ito smaller subproblems solvig (recursively) each subproblem the combiig solutios
More informationChapter 3 Section 3 Lesson Additional Rules for Exponents
Chpter Sectio Lesso Additiol Rules for Epoets Itroductio I this lesso we ll eie soe dditiol rules tht gover the behvior of epoets The rules should be eorized; they will be used ofte i the reiig chpters
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationPermutations, the Parity Theorem, and Determinants
1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits
More informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationLecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a dregular
More informationMATHEMATICS SYLLABUS SECONDARY 7th YEAR
Europe Schools Office of the SecretryGeerl Pedgogicl developmet Uit Ref.: 201101D41e2 Orig.: DE MATHEMATICS SYLLABUS SECONDARY 7th YEAR Stdrd level 5 period/week course Approved y the Joit Techig
More informationTHE LEAST SQUARES REGRESSION LINE and R 2
THE LEAST SQUARES REGRESSION LINE ad R M358K I. Recall from p. 36 that the least squares regressio lie of y o x is the lie that makes the sum of the squares of the vertical distaces of the data poits from
More informationA Gentle Introduction to Algorithms: Part II
A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The BigO, BigΘ, BigΩ otatios: asymptotic bouds
More informationIn this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.
Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More information{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers
. Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,
More informationApplication: Volume. 6.1 Overture. Cylinders
Applictio: Volume 61 Overture I this chpter we preset other pplictio of the defiite itegrl, this time to fid volumes of certi solids As importt s this prticulr pplictio is, more importt is to recogize
More informationEngineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationA Resource for Freestanding Mathematics Qualifications
A pie chrt shows how somethig is divided ito prts  it is good wy of showig the proportio (or frctio) of the dt tht is i ech ctegory. To drw pie chrt:. Fid the totl umer of items.. Fid how my degrees represet
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationPage 2 of 14 = T(2) + 2 = [ T(3)+1 ] + 2 Substitute T(3)+1 for T(2) = T(3) + 3 = [ T(4)+1 ] + 3 Substitute T(4)+1 for T(3) = T(4) + 4 After i
Page 1 of 14 Search C455 Chapter 4  Recursio Tree Documet last modified: 02/09/2012 18:42:34 Uses: Use recursio tree to determie a good asymptotic boud o the recurrece T() = Sum the costs withi each level
More information1 Determinants. Definition 1
Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationOnestep equations. Vocabulary
Review solvig oestep equatios with itegers, fractios, ad decimals. Oestep equatios Vocabulary equatio solve solutio iverse operatio isolate the variable Additio Property of Equality Subtractio Property
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationRoots, Radicals, and Complex Numbers
Chpter 8 Roots, Rils, Comple Numbers Agel, Itermeite Algebr, 7e Lerig Objetives Workig with squre roots Higherorer roots; ris tht oti vribles Simplifig ril epressios Agel, Itermeite Algebr, 7e Squre Roots
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informations = 1 2 at2 + v 0 t + s 0
Mth A UCB, Sprig A. Ogus Solutios for Problem Set 4.9 # 5 The grph of the velocity fuctio of prticle is show i the figure. Sketch the grph of the positio fuctio. Assume s) =. A sketch is give below. Note
More informationRadicals and Fractional Exponents
Radicals ad Roots Radicals ad Fractioal Expoets I math, may problems will ivolve what is called the radical symbol, X is proouced the th root of X, where is or greater, ad X is a positive umber. What it
More informationA. Description: A simple queueing system is shown in Fig. 161. Customers arrive randomly at an average rate of
Queueig Theory INTRODUCTION Queueig theory dels with the study of queues (witig lies). Queues boud i rcticl situtios. The erliest use of queueig theory ws i the desig of telehoe system. Alictios of queueig
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More informationFourier Series and the Wave Equation Part 2
Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries
More informationLecture 11. Shuanglin Shao. October 2nd and 7th, 2013
Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationNUMBERS COMMON TO TWO POLYGONAL SEQUENCES
NUMBERS COMMON TO TWO POLYGONAL SEQUENCES DIANNE SMITH LUCAS Chia Lake, Califoria a iteger, The polygoal sequece (or sequeces of polygoal umbers) of order r (where r is r > 3) may be defied recursively
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More information7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b
Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationMath Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:
Math 355  Discrete Math 4.14.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let
More informationSection 6.1 Radicals and Rational Exponents
Sectio 6.1 Radicals ad Ratioal Expoets Defiitio of Square Root The umber b is a square root of a if b The priciple square root of a positive umber is its positive square root ad we deote this root by usig
More information2.5 Elementary Row Operations and the Determinant
2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)
More informationEquation of a line. Line in coordinate geometry. Slopeintercept form ( 斜 截 式 ) Intercept form ( 截 距 式 ) Pointslope form ( 點 斜 式 )
Chapter : Liear Equatios Chapter Liear Equatios Lie i coordiate geometr I Cartesia coordiate sstems ( 卡 笛 兒 坐 標 系 統 ), a lie ca be represeted b a liear equatio, i.e., a polomial with degree. But before
More informationCHAPTER10 WAVEFUNCTIONS, OBSERVABLES and OPERATORS
Lecture Notes PH 4/5 ECE 598 A. L Ros INTRODUCTION TO QUANTUM MECHANICS CHAPTER0 WAVEFUNCTIONS, OBSERVABLES d OPERATORS 0. Represettios i the sptil d mometum spces 0..A Represettio of the wvefuctio i
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationReleased Assessment Questions, 2015 QUESTIONS
Relesed Assessmet Questios, 15 QUESTIONS Grde 9 Assessmet of Mthemtis Ademi Red the istrutios elow. Alog with this ooklet, mke sure you hve the Aswer Booklet d the Formul Sheet. You my use y spe i this
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationLesson 12. Sequences and Series
Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or
More informationSection 8.3 : De Moivre s Theorem and Applications
The Sectio 8 : De Moivre s Theorem ad Applicatios Let z 1 ad z be complex umbers, where z 1 = r 1, z = r, arg(z 1 ) = θ 1, arg(z ) = θ z 1 = r 1 (cos θ 1 + i si θ 1 ) z = r (cos θ + i si θ ) ad z 1 z =
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationDivide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015
CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationLecture 7: Borel Sets and Lebesgue Measure
EE50: Probability Foudatios for Electrical Egieers JulyNovember 205 Lecture 7: Borel Sets ad Lebesgue Measure Lecturer: Dr. Krisha Jagaatha Scribes: Ravi Kolla, Aseem Sharma, Vishakh Hegde I this lecture,
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More informationHermitian Operators. Eigenvectors of a Hermitian operator. Definition: an operator is said to be Hermitian if it satisfies: A =A
Heriti Opertors Defiitio: opertor is sid to be Heriti if it stisfies: A A Altertively clled self doit I QM we will see tht ll observble properties st be represeted by Heriti opertors Theore: ll eigevles
More informationWe will begin this chapter with a quick refresher of what an exponent is.
.1 Exoets We will egi this chter with quick refresher of wht exoet is. Recll: So, exoet is how we rereset reeted ultilictio. We wt to tke closer look t the exoet. We will egi with wht the roerties re for
More informationCooleyTukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Cosider a legth sequece x[ with a poit DFT X[ where Represet the idices ad as +, +, Cooley CooleyTuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationThe Field of Complex Numbers
The Field of Complex Numbers S. F. Ellermeyer The costructio of the system of complex umbers begis by appedig to the system of real umbers a umber which we call i with the property that i = 1. (Note that
More information1 StateSpace Canonical Forms
StateSpace Caoical Forms For ay give system, there are essetially a ifiite umber of possible state space models that will give the idetical iput/output dyamics Thus, it is desirable to have certai stadardized
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More information