Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix


 Jeffry Robertson
 1 years ago
 Views:
Transcription
1 tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri Rows nd Columns of tri tri ddition Slr ultiplition of tri tri ultiplition rnspose tri he Zero tri 9 he Identity tri he Determinnt nd Inverse of tri he Determinnt of tri he Inverse of tri n lterntive ethod for Clulting the Determinnt of tri Eigenvlues nd Eigenvetors of tri lgerilly we denote mtri with pitl letter, eg numer ourring in mtri is lled n element Eh element in mtri n e identified y its row nd olumn numers For emple the element in position, ) is the element in row nd olumn of the mtri Emple ) For the mtri the element in position, ) is 9 ) he element t lotion, ) is Size of tri We n identify the size dimension) of mtri using numer pir in the form r, where r is the numer of rows in the mtri nd the numer of olumns mtri with the sme numer of rows s olumns is lled squre mtri for ovious resons) Dr Clum donld Emple he mtries in Emple hve the following sizes, ), ), ) nd d)
2 Rows nd Columns of tri nother useful pproh to the struture of mtri is to look t the rows nd olumns of the mtri hese re simple nd ovious onepts; ut we need to know tht the rows re numered strting from the top ie row ) nd the olumns re numered strting from the left hnd side of the mtri ie olumn ) Slr ultiplition of tri mtri n e multiplied y numer nd this proedure is referred to s slr multiplition We perform slr multiplition y multiplying eh element in the mtri y the numer Slr multiplition should not e onfused with mtri multiplition whih will e defined lter nd Row rd Column We n define ddition nd multiplition etween pirs of mtries s long s they re of pproprite dimensions Emple ), ) tri ddition ddition is strightforwrd wo mtries n e dded if nd only if they re of the sme size; the result is hieved y dding orresponding elements of the mtries he result of the ddition is mtri of the sme size Emple ) 9 ) ) 9 w y z w w z y w z 9 9, It is importnt to reognise tht we n not dd mtries with different dimensions For emple, we n not dd mtri to mtri Emple ) Simplify, tri ultiplition tri multiplition n only e rried out etween mtries whih re onformle for mtri multiplition wo mtries nd, with sizes m n nd p q respetively, re onformle for multiplition if nd only if n p ; ie the numer of olumns of is the sme s the numer of rows of he result is written Note tht mtri multiplition my e defined for ut not neessrily for Hene, mtri multiplition is not in generl ommuttive he result of multiplying n m n mtri nd n n q mtri is n m q mtri note the outside size symols give the size of the result) Emple Whih of the following mtries re onformle for mtri multiplition,, C?, C, C nd re vlid multiplitions; wheres we nnot lulte, C, or CC
3 o multiply two mtries, onformle for mtri multiplition, involves n etension of the dot produt proedure desried erlier in the setion on vetors o form the result of multiplying on the left) y on the right) ie to form the produt ) we view s mtri omposed of rows nd s mtri mde up of olumns he entries in the produt mtri re determined y forming dot produts  following the method desried previously o form the i, )th element of we form the dot produt of row i of mtri with olumn of mtri Emple Let nd he produt mtri n e lulted s hs size nd hs size, ie the numer of olumns of is the sme s the numer of rows of Hene, will hve size For emple, to otin the element in position, ) of the produt mtri we tke the dot produt of row of with olumn of, ie o otin the element in position, ) of the produt mtri we tke the dot produt of row of with olumn of, ie in vetor form we hve [ ] his proess n e ontinued to generte the omponents of the produt mtri ) ) ) ) ) ) ) ) Emple 9 Let nd If possile lulte nd In this se we n lulte oth nd Why?) he result of oth the multiplitions will e mtri Why?) nd 9 Note: this is n emple of very importnt result in mtri rithmeti, ie in generl Emple Evlute the following mtri produts: ), ), ) y ), ) ) y y Emple Let nd 9 Clulte the mtri produts nd ), In this se mtri multiplition is ommuttive,, nd this prtiulr produt mtri is lled the identity mtri see Setions 9 nd for further disussions
4 rnspose tri he mtri otined from y interhnging the rows nd the olumns of is lled trnspose nd is denoted Emple Consider the mtri : he mtri Find is otined y interhnging rows nd olumns of the mtri Hene, he Determinnt nd Inverse of tri Consider the following rithmeti evlutions Note tht row of is olumn of nd row of is olumn of One wy of interpreting this lst sttement is tht ny nonzero) numer hs ssoited with it multiplitive inverse, nd tht numer times its inverse equls the identity) We n mke n nlogous sttement for squre mtries tht will prove useful lter on Provided the determinnt defined elow) of squre mtri is non zero, there eists nother squre mtri sme size s ) lled the inverse of nd denoted nd it hs the property tht he Zero tri he zero mtri is mtri for whih every element is zero Stritly there re mny zero mtries one for eh possile size of mtri 9 he Identity tri his refers to the identity with respet to mtri multiplition he identity mtri is only defined for squre mtries: it hs on the min digonl the digonl strting t top left nd going to ottom right) nd zeros everywhere else he mtri is usully represented y I Note tht some tet ooks inlude susript n, nd write I n, to indite the size of the squre mtri Emple ) he identity mtri is, I, ) We met the identity mtri is, ) he identity mtri is, I in Emple, I I where I is similrly sized identity mtri In this se mtri multiplition is ommuttive We now look t how to determine inverse mtries for the se For generl mtri the inverse is simply written down using the formul d d det ) where det ) d is lled the determinnt of the mtri Note tht det ) is often written However, re must e tken not to onfuse this nottion with the modulus We nnot overemphsise the importne of the requirement tht d for the eistene of n inverse of the generl mtri he determinnt of mtri n e used to d determine the eistene, or otherwise, of mtri inverse: y heking tht it is nonzero mtri with n inverse is lled invertile mtri with no inverse is sid to e noninvertile or singulr
5 Emple Clulte the determinnt of eh of the following mtries; hene identify whih mtries re invertile nd for the invertile mtries lulte the inverse ), ), ), he Determinnt of tri We now demonstrte how to lulte the determinnt of mtri Define the mtri o lulte the determinnt we proeed s follows: ) d), e) Determinnt is zero No inverse Epnding y Row : det) det det det Note the sign on the entrl term We do not need to simplify this epression ny further s we lredy know how to lulte the determinnt of mtri lterntively we n epnd on the seond or third rows s follows: ) Determinnt is tri hs n inverse, Epnding y Row : det) det det det ) Determinnt is zero No inverse d) Determinnt is  tri hs n inverse, e) Determinnt is tri hs n inverse, For those mtries for whih n inverse eists you should hek tht I Note: Geometrilly, the solute vlue of the determinnt of mtri is the d re of prllelogrm whose edges re the vetors, ) nd, d) y Epnding y Row : det) det det det In the three ses desried ove note the rry of signs tht prefi the oeffiients i, ie In similr mnner we n epnd on ny of the olumns of using the sign rry ove Epnding y Column : det) det det det Epnding y Column : det) det det det Epnding y Column : det) det det det Note: he vlue of the determinnt will lwys e the sme regrdless of whih row or olumn we perform the epnsion on 9
6 Emple Clulte the determinnt of the mtri 9 : Epnding on the first row gives: Emple det) det det 9 det 9 9 ) 9 ) ) 9 Clulte the determinnt of the mtri : Epnding on the first row gives: det ) det det det ) ) ) s we sw erlier the vlue of the determinnt of squre mtri n e used to determine if mtri is invertile: if the determinnt is nonzero the mtri is invertile; otherwise the mtri is NO invertile Hene, the mtri in Emple is not invertile while the mtri in Emple is invertile provided tht we hve Note: When lulting the determinnt of mtri we usully epnd long the row or olumn ontining most zeros in order to minimize the rithmeti So, in Emple we ould hoose to epnd long row or epnd down olumn he Inverse of tri y the Coftor ethod We now etend the ide of n inverse mtri to the se In generl, the inverse of mtri is given y the formul d ) det ) det ) where the mtri d ) is known s the doint of In order to lulte the inverse of mtri, if it eists, we must therefore otin the determinnt of nd the doint of he derivtion of the doint mtri requires us to lulte quntities known s the minors nd oftors of the mtri he following prgrphs illustrte the methodology y wy of n emple reduing the proedure to five distint steps Emple Determine the inverse of the mtri if it eists SEP : Clulte the determinnt of Epnding long the third row the determinnt of is Sine det ), det ) det )det det eists SEP : Clulte the mtri of minors he minor of entry i, denoted y i, is otined s follows: remove the i th row th remove the olumn the minor i is the determinnt of the remining mtri he minor of entry )
7 he minor of entry he minor of entry he minor of entry he minor of entry ) SEP : Clulte the oftor mtri i he oftor of entry, denoted y C, is defined s C i ) i i o otin the oftor mtri of ) we simply hnge signs of the elements of the mtri of minors lulted in Step using the sign mtri: Hene, we otin the oftor mtri of ) i he minor of entry he minor of entry ) SEP : Clulte the doint mtri he doint of is determined s the trnspose of the oftor mtri, ie d ) he minor of entry he minor of entry Hene, the mtri of minors 9 ) SEP : Clulte the inverse mtri he inverse of the mtri is lulted s d ) det ) so tht whih simplifies to give It is strightforwrd to hek tht I
8 Emple Determine the inverse of the mtri 9 if it eists SEP : Clulte the determinnt of Epnding long the first row the determinnt of is Sine det ), det ) ) eists ) he minor of entry he minor of entry he minor of entry ) SEP : Clulte the mtri of minors he minor of entry i, denoted y i, is otined s follows: remove the i th row th remove the olumn the minor i is the determinnt of the remining mtri he minor of entry he minor of entry he minor of entry he minor of entry Hene, the mtri of minors ) he minor of entry he minor of entry SEP : Clulte the oftor mtri i he oftor of entry, denoted y C, is defined s C i ) i i o otin the oftor mtri of ) we simply hnge signs of the elements of the mtri of minors in Step using the sign mtri: i
9 Hene, we otin the oftor mtri of ) n lterntive ethod for Clulting the Determinnt of tri Rule of Srrus n lterntive pproh for lulting the determinnt of mtri SEP : Clulte the doint mtri he doint of is determined s the trnspose of the oftor mtri, ie SEP : Clulte the inverse mtri he inverse of the mtri is lulted s so tht d ) d ) det ) We will hek tht our inverse is orret s follows: 9 I s required ws developed y the Frenh mthemtiin Pierre Srrus 9) he Rule of Srrus involves the following steps: rewrite the first two olumns of the mtri to the right of it using the left to right digonls tke the produts,, using the right to left digonls tke the produts,, omine the ove nd lulte det ) Note: Geometrilly, the solute vlue of the determinnt of mtri is the volume of prllelepiped whose edges re the vetors u,, ), v,, ) nd w,, ) v w u
10 Emple 9 Clulte the determinnt of the mtri nd find if it eists 9 Emple If nd, then We rewrite s nd lulte 9 det ) Here we hve tht nd so we sy tht is n eigenvetor of orresponding to the eigenvlue λ he geometri effet in this emple is tht the vetor hs een strethed y ftor of ut its diretion remins unhnged s λ > Note tht ny vetor k, where k is onstnt, is n eigenvetor orresponding to the eigenvlue s det ) the mtri is not invertile Geometrilly this result mens tht the vetors,, ),,, ), nd,, 9) re oplnr, ie they lie in the sme plne, so the volume of prllelepiped sed on them is equl to For emple,, eigenvlue λ 9 /,, et re ll eigenvetors orresponding to the Eigenvlues nd Eigenvetors of tri Eigenvlues nd eigenvetors hve mny importnt pplitions in siene nd engineering inluding solving systems of differentil equtions, stility nlysis, virtion nlysis nd modelling popultion dynmis Let e n n) mtri n eigenvlue of is slr rel or omple) suh tht I) for some nonzero vetor In this se, we ll the vetor n eigenvetor of orresponding to the eigenvlue Geometrilly Eq I) mens tht the vetors nd re prllel he vlue of determines wht hppens to when it is multiplied y, ie whether it is shrunk or strethed or if its diretion is unhnged or reversed Clultion of Eigenvlues If is ) or ) mtri it is usully reltively strightforwrd to lulte its eigenvlues nd eigenvetors y hnd So, how do we lulte them? We hve tht I, where I is the identity mtri, so we n rewrite Eq I) s I  I λ I) We note tht the eqution λ I) n only hold for nonzero vetor if the mtri λ I) is singulr does not hve n inverse) Hene, the eigenvlues of re the numers for whih the mtri λ I) does not hve n inverse In other words the numers stisfy the eqution det λ I) II) nd, s noted ove, they n e rel or omple 9
11 Emple Find the eigenvlues of the following mtries i) ii) iii) C s λ i) λ I λ λ λ Hene, det λ I) λ) λ) ) ) λ λ λ We ll λ λ the hrteristi polynomil of the mtri nd the eigenvlues of stisfy the hrteristi eqution det λ I), ie λ λ λ ) λ ) λ nd λ Hene, λ nd λ re the eigenvlues of the mtri Note: We n esily hek our nswer s follows: Let tr) denote the tre of mtri, ie the sum of the elements on the min digonl hen the sum of the eigenvlues equls the tre of the mtri Here we hve tht tr) nd the sum of the eigenvlues is s required λ iii) C λ I λ λ λ Hene, det C λ I) λ) λ) )) λ λ Now solve det C λ I) to find the eigenvlues of C, ie λ λ ± Hene, the eigenvlues of the mtri C re λ nd λ he following emple demonstrtes shortut pproh tht n e dopted when lulting the eigenvlues of speifi types of mtries Emple Find the eigenvlues of the following mtries i) ii) iii) C s We first lulte the eigenvlues using the method desried ove efore identifying shortut pproh for these speil types of mtries λ i) Solving det λ I) gives λ λ ) λ) λ, λ λ ii) λ I λ λ λ Hene, det λ I) λ) λ) ) ) λ λ λ Now solve det λ I) to find the eigenvlues of, ie λ λ λ ) λ ) λ nd λ λ ii) Solving det λ I) gives λ λ iii) Solving det C λ I) gives λ λ ) λ) λ, λ λ ) λ) λ, λ Hene, λ nd λ re the eigenvlues of the mtri
12 In this emple: tri is lowertringulr mtri nd hs the property tht ll of its entries ove the min digonl re tri is n uppertringulr mtri nd hs the property tht ll of its entries elow the min digonl re tri C is digonl mtri nd hs the property tht ll of its entries not on the min digonl re Note here tht it is possile to hve some of the entries on the min digonl equl to zero Shortut: In ll three ses  lower tringulr, uppertringulr nd digonl  the eigenvlues re simply the entries on the min digonl nd so we n ust red them off without the need for ny lultions Notes For n n) mtri the hrteristi eqution is polynomil of degree n In the emples ove we sw tht the hrteristi eqution in the se is qudrti he sum of the eigenvlues is equl to the tre of the mtri he produt of the eigenvlues is equl to the determinnt of the mtri If is n eigenvlue of mtri then is not invertile If is n eigenvlue of n invertile mtri then is n eigenvlue of λ he mtri nd its trnspose,, hve the sme eigenvlues Repeted Eigenvlues In the emples presented up to now the eigenvlues hve een distint ut it is possile for mtri to hve repeted eigenvlues Emple Find the eigenvlues of the mtri o find the eigenvlues we solve 9 9 λ 9 det λ I) 9 λ λ )9 λ) 9 λ λ λ ) λ ) λ repeted) Comple Eigenvlues It is possile for relvlued mtri to hve omple eigenvlues nd eigenvetors) s illustrted y the following emple Emple Find the eigenvlues of the mtri o find the eigenvlues we solve λ det λ I) λ λ λ ±
13 Clultion of Eigenvetors One we hve lulted the eigenvlues we n find the eigenvetors y solving the mtri eqution for eh eigenvlue in turn Note here tht eigenvetors must e nonzero he eqution given ove is often written s Emple λ I) Find the eigenvlues nd eigenvetors of the mtri First we find the eigenvlues y solving: λ det λ I) λ λ ) λ) λ λ λ ) λ ) λ, λ We now lulte the eigenvetors orresponding to the eigenvlues y solving the eigenvetor eqution, Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ hese re simultneous equtions nd we note tht one eqution must e multiple of the other If not then you hve mde mistke! Here Eq ) is times Eq ) oth the equtions give nd if we let α, sy, for some nonzero numer α, then α nd we find the first eigenvetor to e of the form α α α Note tht there re infinitely mny nonzero eigenvetors depending on the vlue hosen for α Setting α gives n eigenvetor orresponding to the eigenvlue λ s We n hek our nswer y showing tht λ We hve Hene, λ s required nd λ Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ oth these equtions give Let α, α ) then α nd so ) ) α α α
14 Setting α gives Hene, λ s required It is reltively strightforwrd to hek tht, ie nd λ In summry, we therefore hve the eigenvlue/eigenvetor pirs, Emple λ, ; λ, Find the eigenvlues nd eigenvetors of the following mtri : In Emple prt ii) we found the eigenvlues of to e λ nd λ We now lulte the eigenvetors orresponding to these eigenvlues y solving the eigenvetor eqution, Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ oth these equtions give Note tht for system we do not tully need to introdue the prmeter s we did in the previous emple We n simply hoose onvenient numeril vlue for either of the omponents or of the eigenvetor So here we n let, sy, giving hen n eigenvetor orresponding to the eigenvlue λ is Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ oth these equtions give Let, sy, so tht hen n eigenvetor orresponding to the eigenvlue λ is In summry, we therefore hve the eigenvlue/eigenvetor pirs, λ, ; λ,
15 Systems With Zero s n Eigenvlue We hve previously noted tht n eigenvetor nnot e the zero vetor,, ut it is possile to hve n eigenvlue λ Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ Emple Find the eigenvlues nd eigenvetors of the mtri o find the eigenvlues we need to solve λ det λ I) λ λ ) λ) λ λ λ λ ) λ, λ We now find the eigenvetors orresponding to these eigenvlues: Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ oth these equtions give Let, sy, so tht n eigenvetor orresponding to the eigenvlue λ will then e oth these equtions give Let, sy, then n eigenvetor orresponding to the eigenvlue λ will then e o summrise we hve: λ, ; λ, Emple Find the eigenvetors of the mtri in Emple We previously found tht hd omple eigenvlues, Cse : o find n eigenvetor orresponding to eigenvlue λ λ nd λ λ we solve 9
16 If, for emple, we multiply the first eqution y oth equtions give Let, sy, then n eigenvetor orresponding to the eigenvlue λ will then e Cse : o find n eigenvetor orresponding to eigenvlue λ we solve λ If, for emple, we multiply the seond eqution y oth equtions give Let, sy, then n eigenvetor orresponding to the eigenvlue λ will then e o summrise we hve: λ, ; λ, utoril Eerises ) Simplify the following i) ii) iii) iv) ) Simplify the following mtri produts i) ii) iii) [ ] iv) [ ] ) Simplify the following nd omment on your nswers i) ii) ) i) Whih of the following mtries n e squred?, ii) In generl, whih mtries n e squred?
17 ) i) Given tht, find the inverse mtri nd lulte the mtri produts nd Comment on your results ) Given tht nd, find the inverse mtri nd lulte the mtri produts ii) Given tht lulte the mtri produt ) Determine when the following mtri is invertile nd lulte its inverse k ) For the mtri show tht ) ) Given tht D, find the inverse mtri D ) Consider the mtries D nd P mtri P nd lulte the mtri produt P D P Determine the inverse ) Let nd Evlute ) nd Comment on your nswer nd ) Find the eigenvlues of eh of the following mtries: i) ii) iii) iv) 9) Let nd Evlute ) nd Comment on your nswer ) Evlute the determinnt of eh of the following mtries ) Find the eigenvlues nd eigenvetors of eh of the following mtries: i) ii) iii) iv) i) ii) iii) ) Find the eigenvlues nd eigenvetors of eh of the following mtries: ) Whih of the mtries in Question re invertile? Justify your nswer i) ii) iii) iv) ) For eh invertile mtri in Question determine the inverse
18 nswers ) i) ; ii), iii) ; iv) ) i) ; ii) ; iii) 9 iv) ) i) ii) he onlusion from this emple is tht mtri multiplition is not ommuttive, so tht the order in whih mtries re multiplied is importnt ) i) Only the first mtri n e squred sine it is onformle for multiplition with itself ii) In generl, to squre mtri of size p m requires multiplying n p m mtri y n p m mtri hese re only onformle for mtri multiplition if p m ) i) so det) whih is nonzero nd so the mtri is invertile hen We hve tht I oth mtri produts give the identity mtri ii) so nd 9 ) k so k ) det his is nonzero provided k nd in this se the inverse is k k ) ) s required ) so ) In generl for mtries nd we hve tht ) 9) nd so ) lso, nd so ) In generl for n n squre mtries nd we hve tht ) ) i) det det det D ii) det det det D
19 iii) D det det det ) i) λ, λ ii) λ, λ iii) λ, λ iv) λ, λ ) hey re ll invertile eept iii), sine only iii) hs zero determinnt ) i) ii) ) i) λ, ii) λ, λ, ; λ, ) iii) λ, ; λ, iv) λ, ; λ, ) D ) i) λ, ; λ, Note: If D is n n n n digonl mtri then its inverse is given y ii) λ, iii) λ, ; λ, ; λ, D n n provided tht none of the digonl elements re zero iv) λ, ; λ, ) P nd P D P
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose
More information1.3 SCALARS AND VECTORS
Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd
More informationProject 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More informationQUADRATIC EQUATION. Contents
QUADRATIC EQUATION Contents Topi Pge No. Theory 004 Exerise  0509 Exerise  093 Exerise  3 45 Exerise  4 6 Answer Key 78 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,
More informationChapter Gauss Quadrature Rule of Integration
Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to
More information6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
More informationLesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.
27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationPythagoras theorem and surds
HPTER Mesurement nd Geometry Pythgors theorem nd surds In IEEM Mthemtis Yer 8, you lernt out the remrkle reltionship etween the lengths of the sides of rightngled tringle. This result is known s Pythgors
More informationNONDETERMINISTIC FSA
Tw o types of nondeterminism: NONDETERMINISTIC FS () Multiple strtsttes; strtsttes S Q. The lnguge L(M) ={x:x tkes M from some strtstte to some finlstte nd ll of x is proessed}. The string x = is
More informationChapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
More informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd AmGm inequlity 2. Elementry inequlities......................
More informationThis enables us to also express rational numbers other than natural numbers, for example:
Overview Study Mteril Business Mthemtis 0506 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers
More informationMT Integral equations
MT58  Integrl equtions Introduction Integrl equtions occur in vriety of pplictions, often eing otined from differentil eqution. The reson for doing this is tht it my mke solution of the prolem esier or,
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More informationSection 4.4. Green s Theorem
The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higherdimensionl nlogues) with the definite integrls
More informationThe RiemannStieltjes Integral
Chpter 6 The RiemnnStieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0
More informationChapter 2. Determinants
Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if dbc0. The expression dbc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is
More informationLecture 1  Introduction and Basic Facts about PDEs
* 18.15  Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1  Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV
More informationVectors. Chapter14. Syllabus reference: 4.1, 4.2, 4.5 Contents:
hpter Vetors Syllus referene:.,.,.5 ontents: D E F G H I J K Vetors nd slrs Geometri opertions with vetors Vetors in the plne The mgnitude of vetor Opertions with plne vetors The vetor etween two points
More informationm A 1 1 A ! and AC 6
REVIEW SET A Using sle of m represents units, sketh vetor to represent: NONCALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in northesterl diretion. Simplif:
More informationILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS
ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu
More informationLine Integrals and Entire Functions
Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series
More informationCore 2 Logarithms and exponentials. Section 1: Introduction to logarithms
Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI
More informationu(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.
Lecture 4 Complex Integrtion MATHGA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationDeterminants Chapter 3
Determinnts hpter Specil se : x Mtrix Definition : the determinnt is sclr quntity defined for ny squre n x n mtrix nd denoted y or det(). x se ecll : this expression ppers in the formul for x mtrix inverse!
More informationMatrix Eigenvalues and Eigenvectors September 13, 2017
Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues
More informationSolving Radical Equations
Solving dil Equtions Equtions with dils: A rdil eqution is n eqution in whih vrible ppers in one or more rdinds. Some emples o rdil equtions re: Solution o dil Eqution: The solution o rdil eqution is the
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More information9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes
The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More information8.3 THE HYPERBOLA OBJECTIVES
8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS
More informationm m m m m m m m P m P m ( ) m m P( ) ( ). The oordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r
COORDINTE GEOMETR II I Qudrnt Qudrnt (.+) (++) X X    0  III IV Qudrnt  Qudrnt ()  (+) Region CRTESIN COORDINTE SSTEM : Retngulr Coordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr
More information(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.
Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time
More informationProving the Pythagorean Theorem
Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or
More informationSolutions to Assignment 1
MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove
More informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More informationDETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ
All Mthemticl truths re reltive nd conditionl. C.P. STEINMETZ 4. Introduction DETERMINANTS In the previous chpter, we hve studied bout mtrices nd lgebr of mtrices. We hve lso lernt tht system of lgebric
More informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors
More informationdx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.
Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd
More informationA Nonparametric Approach in Testing Higher Order Interactions
A Nonprmetri Approh in Testing igher Order Intertions G. Bkeerthn Deprtment of Mthemtis, Fulty of Siene Estern University, Chenkldy, Sri Lnk nd S. Smit Deprtment of Crop Siene, Fulty of Agriulture University
More informationMath 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)
Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte singlevrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion
More informationCSCI 5525 Machine Learning
CSCI 555 Mchine Lerning Some Deini*ons Qudrtic Form : nn squre mtri R n n : n vector R n the qudrtic orm: It is sclr vlue. We oten implicitly ssume tht is symmetric since / / I we write it s the elements
More informationQUADRATIC EQUATION EXERCISE  01 CHECK YOUR GRASP
QUADRATIC EQUATION EXERCISE  0 CHECK YOUR GRASP. Sine sum of oeffiients 0. Hint : It's one root is nd other root is 8 nd 5 5. tn other root 9. q 4p 0 q p q p, q 4 p,,, 4 Hene 7 vlues of (p, q) 7 equtions
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationalong the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate
L8 VECTOR EQUATIONS OF LINES HL Mth  Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne
More informationCalculus Cheat Sheet. Integrals Definitions. where F( x ) is an antiderivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx
Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous AntiDerivtive : An ntiderivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.
More informationare coplanar. ˆ ˆ ˆ and iˆ
SMLE QUESTION ER Clss XII Mthemtis Time llowed: hrs Mimum Mrks: Generl Instrutions: i ll questions re ompulsor. ii The question pper onsists of 6 questions divided into three Setions, B nd C. iii Question
More informationMore Properties of the Riemann Integral
More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl
More informationBasic Angle Rules 5. A Short Hand Geometric Reasons. B Two Reasons. 1 Write in full the meaning of these short hand geometric reasons.
si ngle Rules 5 6 Short Hnd Geometri Resons 1 Write in full the mening of these short hnd geometri resons. Short Hnd Reson Full Mening ) se s isos Δ re =. ) orr s // lines re =. ) sum s t pt = 360. d)
More informationNon Right Angled Triangles
Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in nonright ngled tringles. This unit
More informationH (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.
Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining
More informationFunctions. mjarrar Watch this lecture and download the slides
9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. OnetoOne Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides
More informationNaming the sides of a rightangled triangle
6.2 Wht is trigonometry? The word trigonometry is derived from the Greek words trigonon (tringle) nd metron (mesurement). Thus, it literlly mens to mesure tringle. Trigonometry dels with the reltionship
More informationBoolean Algebra. Boolean Algebra
Boolen Alger Boolen Alger A Boolen lger is set B of vlues together with:  two inry opertions, commonly denoted y + nd,  unry opertion, usully denoted y ˉ or ~ or,  two elements usully clled zero nd
More informationSolutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!
Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd GussJordn elimintion to solve systems of liner
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More information( ) as a fraction. Determine location of the highest
AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to
More informationAnalytically, vectors will be represented by lowercase boldface Latin letters, e.g. a, r, q.
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
More informationwhere the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b
CS 2942 9/11/04 Quntum Ciruit Model, SolovyKitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits  Universl Gte Sets A lssil iruit implements multioutput oolen funtion f : {0,1}
More informationECON 331 Lecture Notes: Ch 4 and Ch 5
Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationfor all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx
Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion
More informationA Mathematical Model for UnemploymentTaking an Action without Delay
Advnes in Dynmil Systems nd Applitions. ISSN 97353 Volume Number (7) pp. 8 Reserh Indi Publitions http://www.ripublition.om A Mthemtil Model for UnemploymentTking n Ation without Dely Gulbnu Pthn Diretorte
More informationThe Ellipse. is larger than the other.
The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)
More informationPythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides
Pythgors theorem nd trigonometry Pythgors Theorem The hypotenuse of rightngled tringle is the longest side The hypotenuse is lwys opposite the rightngle 2 = 2 + 2 or 2 = 22 or 2 = 22 The re of the
More informationLesson Notes: Week 40Vectors
Lesson Notes: Week 40Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationMath 259 Winter Solutions to Homework #9
Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More informationChapter 4 StateSpace Planning
Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 StteSpe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different
More informationENERGY AND PACKING. Outline: MATERIALS AND PACKING. Crystal Structure
EERGY AD PACKIG Outline: Crstlline versus morphous strutures Crstl struture  Unit ell  Coordintion numer  Atomi pking ftor Crstl sstems on dense, rndom pking Dense, regulr pking tpil neighor ond energ
More informationSUBJECT: MATHEMATICS CLASS :XII
SUBJECT: MATHEMATICS CLASS :XII KENDRIYA VIDYALAYA SANGATHAN REGIONAL OFFICE CHANDIGARH YEAR 00 INDEX Sl. No Topis Pge No.. Detil of the onepts 4. Reltions & Funtions 9. Inverse Trigonometri Funtions
More informationDynamics of Structures
UNION Dymis of Strutures Prt Zbigiew Wójii Je Grosel Projet ofied by Europe Uio withi Europe Soil Fud UNION Mtries Defiitio of mtri mtri is set of umbers or lgebri epressios rrged i retgulr form with
More information1 Nondeterministic Finite Automata
1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you
More informationT b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.
Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene
More informationLecture 2 : Propositions DRAFT
CS/Mth 240: Introduction to Discrete Mthemtics 1/20/2010 Lecture 2 : Propositions Instructor: Dieter vn Melkeeek Scrie: Dlior Zelený DRAFT Lst time we nlyzed vrious mze solving lgorithms in order to illustrte
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4305 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system
More informationAn introduction to groups
n introdution to groups syllusref efereneene ore topi: Introdution to groups In this h hpter Groups The terminology of groups Properties of groups Further exmples of groups trnsformtions 66 Mths Quest
More informationMath 017. Materials With Exercises
Mth 07 Mterils With Eercises Jul 0 TABLE OF CONTENTS Lesson Vriles nd lgeric epressions; Evlution of lgeric epressions... Lesson Algeric epressions nd their evlutions; Order of opertions....... Lesson
More informationThe Word Problem in Quandles
The Word Prolem in Qundles Benjmin Fish Advisor: Ren Levitt April 5, 2013 1 1 Introdution A word over n lger A is finite sequene of elements of A, prentheses, nd opertions of A defined reursively: Given
More informationGénération aléatoire uniforme pour les réseaux d automates
Génértion létoire uniforme pour les réseux d utomtes Niols Bsset (Trvil ommun ve Mihèle Sori et Jen Miresse) Université lire de Bruxelles Journées Alé 2017 1/25 Motivtions (1/2) p q Automt re omnipresent
More information6.1 Definition of the Riemann Integral
6 The Riemnn Integrl 6. Deinition o the Riemnn Integrl Deinition 6.. Given n intervl [, b] with < b, prtition P o [, b] is inite set o points {x, x,..., x n } [, b], lled grid points, suh tht x =, x n
More informationLIP. Laboratoire de l Informatique du Parallélisme. Ecole Normale Supérieure de Lyon
LIP Lortoire de l Informtique du Prllélisme Eole Normle Supérieure de Lyon Institut IMAG Unité de reherhe ssoiée u CNRS n 1398 Onewy Cellulr Automt on Cyley Grphs Zsuzsnn Rok Mrs 1993 Reserh Report N
More informationReflection Property of a Hyperbola
Refletion Propert of Hperol Prefe The purpose of this pper is to prove nltill nd to illustrte geometrill the propert of hperol wherein r whih emntes outside the onvit of the hperol, tht is, etween the
More informationMATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC
FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot
More informationCS 310 (sec 20)  Winter Final Exam (solutions) SOLUTIONS
CS 310 (sec 20)  Winter 2003  Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)
More informationCS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation
CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes
More informationThermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R
/10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?
More informationf (x)dx = f(b) f(a). a b f (x)dx is the limit of sums
Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx
More informationPreLie algebras, rooted trees and related algebraic structures
PreLie lgers, rooted trees nd relted lgeri strutures Mrh 23, 2004 Definition 1 A prelie lger is vetor spe W with mp : W W W suh tht (x y) z x (y z) = (x z) y x (z y). (1) Exmple 2 All ssoitive lgers
More informationCS 311 Homework 3 due 16:30, Thursday, 14 th October 2010
CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w
More informationBIFURCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS
BIFRCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible
More informationThe Fundamental Theorem of Algebra
The Fundmentl Theorem of Alger Jeremy J. Fries In prtil fulfillment of the requirements for the Mster of Arts in Teching with Speciliztion in the Teching of Middle Level Mthemtics in the Deprtment of Mthemtics.
More information5.5 The Substitution Rule
5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n ntiderivtive is not esily recognizble, then we re in
More information