Addendum. Addendum. Vector Review. Department of Computer Science and Engineering 1-1

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1 Addedum Addedum Vetor Review Deprtmet of Computer Siee d Egieerig -

2 Coordite Systems Right hded oordite system Addedum y z Deprtmet of Computer Siee d Egieerig -

3 -3 Deprtmet of Computer Siee d Egieerig Addedum Vetor Arithmeti [ ] [ ] [ ] [ ] [ ] [ ] z y z y z z y y z z y y z y z y s s s s

4 Vetor gitude The mgitude legth of vetor is: Addedum v y v + v + v z A vetor with legth. is lled uit vetor We lso ormlize vetor to mke it uit vetor: v v Deprtmet of Computer Siee d Egieerig -4

5 Dot Produt Addedum i i + y y + z z osθ Deprtmet of Computer Siee d Egieerig -5

6 -6 Deprtmet of Computer Siee d Egieerig Addedum Dot Produt T z z y y i i + + osθ [ ] z y z y

7 Emple: Agle Betwee Vetors Addedum How do you fid the gle θ etwee vetors d? θ Deprtmet of Computer Siee d Egieerig -7

8 -8 Deprtmet of Computer Siee d Egieerig Addedum Emple: Agle Betwee Vetors os os os θ θ θ θ

9 Dot Produts with Geerl Vetors Addedum The dot produt is slr vlue tht tells us somethig out the reltioship etwee two vetors If > the θ < 9º If < the θ > 9º If the θ 9º or oe or more of the vetors is degeerte,, Deprtmet of Computer Siee d Egieerig -9

10 Dot Produts with Oe Uit Vetor Addedum If u. the u is the legth of the projetio of oto u u u Deprtmet of Computer Siee d Egieerig -

11 Emple: Diste to Ple Addedum A ple is desried y poit p o the ple d uit orml. Fid the diste from poit to the ple p Deprtmet of Computer Siee d Egieerig -

12 Emple: Diste to Ple Addedum The diste is the legth of the projetio of -p oto : dist p -p p Deprtmet of Computer Siee d Egieerig -

13 Dot Produts with Uit Vetors Addedum < < θ - < <. - os θ Deprtmet of Computer Siee d Egieerig -3

14 -4 Deprtmet of Computer Siee d Egieerig Addedum [ ] y y z z y z z y z y z y k j i Cross Produt

15 Properties of the Cross Produt Addedum is vetor perpediulr to oth d, i the diretio defied y the right hd rule si θ re of prllelogrm if d re prllel Deprtmet of Computer Siee d Egieerig -5

16 Emple: Norml of Trigle Addedum Fid the uit legth orml of the trigle defied y 3D poits,, d Deprtmet of Computer Siee d Egieerig -6

17 Emple: Norml of Trigle Addedum - - Deprtmet of Computer Siee d Egieerig -7

18 Emple: Are of Trigle Fid the re of the trigle defied y 3D poits,, d Addedum Deprtmet of Computer Siee d Egieerig -8

19 Emple: Are of Trigle Addedum re - - Deprtmet of Computer Siee d Egieerig -9

20 Emple: Aligmet to Trget Addedum A ojet is t positio p with uit legth hedig of h. We wt to rotte it so tht the hedig is fig some trget t. Fid uit is d gle θ to rotte roud. t p h Deprtmet of Computer Siee d Egieerig -

21 Emple: Aligmet to Trget Addedum h h t p t p t-p t θ os h t p t p p h θ Deprtmet of Computer Siee d Egieerig -

22 Addedum Addedum Crmer s Rule Deprtmet of Computer Siee d Egieerig -

23 This is gret method. Addedum Crmer s Rule is et wy to evlute systems d if you put the work i ow you ll do fie. It e used for y size y, 3 y 3 or eve lrger system. It is esy to memorize d fst. I m goig to show you where Crmer s Rule omes from; ut first, some defiitios Deprtmet of Computer Siee d Egieerig -3

24 Defiitios Determit squre rry Addedum d Order Determit y rry 3 rd Order Determit 3 y 3 rry Elemets The thigs i the rry Deprtmet of Computer Siee d Egieerig -4

25 Wht does determit look like? A d order determit looks like this Addedum Ad the vlue of the determit d e Digol dow right digol dow left d e e d Deprtmet of Computer Siee d Egieerig -5

26 Emples Evlute Addedum Deprtmet of Computer Siee d Egieerig -6

27 Why is this useful for systems? Addedum ets work through elimitio emple usig ll vriles; the we see how the determit will e useful i solvig. Deprtmet of Computer Siee d Egieerig -7

28 Addedum + y d + ey f ets elimite y e + ey y e d + ey f e d e f e d e f e f e d ook fmilir? f d e e Deprtmet of Computer Siee d Egieerig -8

29 Addedum If you pply the sme proess ut elimite y f e d d y d f d e So, wht does Crmer s Rule sy? Deprtmet of Computer Siee d Egieerig -9

30 Crmer s Rule + y Give system d + ey f Reple solutios i olum to solve for f d e e Wht do you thik is the trik? y d d f e Addedum Reple solutios i y olum to solve for y Deomitors re oeffiiet determits Deprtmet of Computer Siee d Egieerig -3

31 Emples Solve usig Crmer s Rule Addedum y 9 y Deprtmet of Computer Siee d Egieerig -3

32 Crmer s Rule i Geerl Addedum Crmer's Rule: For the system of equtios A y, where A is osigulr mtri, the solutio for the ith edogeous vrile, i, is i A i / A where the mtri A i represets mtri tht is idetil to the mtri A ut for the replemet of the ith olum with the vetor y. Deprtmet of Computer Siee d Egieerig -3

33 -33 Deprtmet of Computer Siee d Egieerig Addedum ier Systems i tri Form O

34 Solutio of ier Systems Eh side of the equtio Addedum A C e multiplied y A - : A A A Due to the defiitio of A - : A A I Therefore the solutio of is: A Deprtmet of Computer Siee d Egieerig -34

35 Cosistey Solvility A - does ot eist for every A Addedum The lier system of equtios A hs solutio, or sid to e osistet IFF Rk{A}Rk{A } A system is iosistet whe Rk{A}<Rk{A } Rk{A} is the mimum umer of lierly idepedet olums or rows of A. Rk e foud y usig ERO Elemetry Row Oprtios or ECO Elemetry olum opertios. Deprtmet of Computer Siee d Egieerig -35

36 Elemetry row d olum opertios th for CS eture 36 Deprtmet of Computer Siee d Egieerig Addedum The followig opertios pplied to the ugmeted mtri [A ], yield equivlet lier system Iterhges: The order of two rows/olums e hged Slig: ultiplyig row/olum y ozero ostt Sum: The row e repled y the sum of tht row d ozero multiple of y other row. Oe use ERO d ECO to fid the Rk s follows: ERO miimum # of rows with t lest oe ozero etry or ECO miimum # of olums with t lest oe ozero etry -36

37 -37 Deprtmet of Computer Siee d Egieerig Addedum A iosistet emple: Geometri iterprettio ERO:ultiply the first row with - d dd to the seod row Rk{A} Rk{A } > Rk{A} 3 4

38 Uiqueess of solutios Addedum The system hs uique solutio IFF Rk{A}Rk{A } is the order of the system Suh systems re lled full-rk systems Deprtmet of Computer Siee d Egieerig -38

39 -39 Deprtmet of Computer Siee d Egieerig Addedum Full-rk systems If Rk{A} Det{A} A - eists Uique solutio 4

40 Rk defiiet mtries If Rk{A}m< Det{A} A is sigulr so ot ivertile ifiite umer of solutios -m free vriles uder-determied system Addedum th for Cosistet CS so solvle Rk{A}Rk{A } Deprtmet of Computer Siee d Egieerig -4

41 -4 Deprtmet of Computer Siee d Egieerig Addedum Ill-oditioed system of equtios A smll devitio i the etries of A mtri, uses lrge devitio i the solutio

42 Ill-oditioed otiued... Addedum A lier system of equtios is sid to e ill-oditioed if the oeffiiet mtri teds to e sigulr th for CS eture 4 Deprtmet of Computer Siee d Egieerig -4

43 -43 Deprtmet of Computer Siee d Egieerig Addedum Gussi Elimitio By usig ERO, mtri A is trsformed ito upper trigulr mtri ll elemets elow digol Bk sustitutio is used to solve the upper-trigulr system i i i i ii i i O O ERO i i i ii i ~ ~ ~ ~ ~ O O Bk sustitutio

44 Pivotl Elemet Addedum Pivotl elemet O The first oeffiiet of the first row pivot is used to zero out first oeffiiets of other rows. I terms of umeril stility it is usully est to use the lrgest elemet i the olum, i.e. m k j k jk th for CS eture Deprtmet of Computer Siee d Egieerig -44

45 -45 Deprtmet of Computer Siee d Egieerig Addedum First step of elimitio , 3, 3, / / / m m m O First row, multiplied y pproprite ftor is sutrted from other rows.

46 -46 Deprtmet of Computer Siee d Egieerig Addedum th for CS Seod step of elimitio O Pivotl elemet The seod oeffiiet pivot of the seod row is used to zero out seod oeffiiets of other rows.

47 -47 Deprtmet of Computer Siee d Egieerig Addedum Seod step of elimitio , 3 3, / / m m O Seod row, multiplied y pproprite ftor is sutrted from other rows ERO.

48 Gussio elimitio lgorithm Addedum Defie umer of steps s p pivotl row For p,- For rp+ to p p m / r, p p rp rp pp m p+ p p r r r, p p For p+ to p+ p m p r r r, p p Deprtmet of Computer Siee d Egieerig -48

49 -49 Deprtmet of Computer Siee d Egieerig Addedum Bk sustitutio lgorithm O Filly, the followig system is otied:

50 -5 Deprtmet of Computer Siee d Egieerig Addedum Bk sustitutio lgorithm [ ],,, K + i i k k i ik i i i ii i The swer is otied s followig:

51 -5 Deprtmet of Computer Siee d Egieerig Addedum U deompositio R O If we ow defie mtri R d other mtri l ij with k kk k ik ik l the we get: A R

52 U deompositio Addedum Note tht the mtries R d re upper d lower trigulr mtries. Hee, we solve the lier equtio system i two steps: R Solvig these two systems e hieved similr to the usig similr lgorithm we used for k sustitutio. Also ote tht solvig the system of lier equtios for differet solutios does ot require repetitio of the Gussi elimitio lgorithm. Deprtmet of Computer Siee d Egieerig -5

53 Cholesky deompositio If the mtri A is symmetri, solutio for the U deompositio is eve esier sie there eist the followig deompositio for those mtries: where A C T C Addedum C O O O Deprtmet of Computer Siee d Egieerig -53

54 -54 Deprtmet of Computer Siee d Egieerig Addedum Cholesky deompositio How we ompute C? et us look t the equtio: Hee: O O O O O O l k kk k j jl jk kl kl k j jl jk kl k j jk kk kk k j jk jk kk < for

ECE 102 Engineering Computation

ECE 102 Engineering Computation ECE Egieerig Computtio Phillip Wog Mth Review Vetor Bsis Mtri Bsis System of Lier Equtios Summtio Symol is the symol for summtio. Emple: N k N... 9 k k k k k the, If e e e f e f k Vetor Bsis A vetor is