ECE 102 Engineering Computation

Transcription

1 ECE Egieerig Computtio Phillip Wog Mth Review Vetor Bsis Mtri Bsis System of Lier Equtios

2 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

3 Vetor Bsis A vetor is set of umeri or symoli elemets tht re rrged i the form of list. Row vetor oe row d olums Emple: [.7 ] so Colum vetor oe olum d mrows Emple: so m

4 Addig d sutrtig two vetors Both vetors must e the sme type d size. [ ] A Colum [ ] B [ ] ± ± ± A ± B ide C Row ide d d d ± ± ± ± d d d C

5 Emple: [ ] [ ] C B A B A [ ] [ ] A B C 7 C [ ] [ ]

6 Multiplyig vetor y slr vlue Eh elemet of the vetor is multiplied y the slr vlue. [ ] A [ ] A B B

7 Emple: [ ] [ ] 8 C B A [ ] [ ] A 8 9 C [ ] [ ] 8 8 B

8 "Multiplyig" two vetors together The ier dot produt of two vetors is slr. [ ] L A [ ] L B j j K B A 7 m M C d m d d M j m k k d k C

9 Emple: A C [ ] B [ 8 ] A B 8 C 8

10 The ross produt of two -dimesiol vetors is other -dimesiol vetor. A B B A 9 Emple: 7 h g h g

11 Mtri Bsis A mtri is set of umeri or symoli elemets rrged i the form of tle. Mtries re ofte used to desrie lier equtios d to perform lier trsformtios. Mtri of dimesios m mrows, olums. 7. Emple: so d. m

12 The vetor s mtri Suppose mtri hs m rows d olums: If m, the the mtri is of size, whih is equivlet to row vetor. [ ] [ K ] K If, the the mtri is of size m, whih is equivlet to olum vetor. M M m m

13 Addig d sutrtig two mtries Both mtries must hve the sme sizes. B A Row ide Colum ide ± ± ± ± ± ± ± ± ± ± B A

14 Emple: 7 8 C B A 8 8 B A C

15 Multiplyig mtri y slr vlue Eh elemet of the mtri is multiplied y the slr vlue. A A

16 Multiplyig two mtries together Give: Mtri Aof size m with elemets Mtri Bof size m with elemets If m, the the produt mtri ABeists d is of size m. The vlue i row id olum jof the produt is: AB ij k ik kj for i m d j

17 To multiply two mtries, the umer of olums of the first mtri must equl the umer of rows of the seod mtri. Multiplyig two mtries is doe y multiplyig oseutive rows from the first mtri y olums from the seod mtri. Mtri multiplitio is usully otommuttive, i.e., AB BAeept for erti Ad B.

18 A B Let C AB Fid : Fid : C 7

19 A B Let C AB Fid : Fid : Fid : Fid : 8

20 Emple: H B A AH 9 8 AB BA

21 A B AB A ol of row of A olof B row of B row of A ol of B row of A olof B row of A ol of B row of A ol of B row of A olof B row of A ol of B row of A ol of B AB

22 A B AB

23 Squre mtri If the umer of olums is equl to the umer of rows i.e., m, the mtri is lled squre. The digol from the upper left orer to the lower right orer is lled the mi digol. Emple: squre mtri 9. Mi digol

24 etermit A determit is slr vlue ssoited with squre mtri y the followig equtios: : A det d A A d : B d e f g h i det d Red sig Blue sig B B ei fg dh fh di eg d g e h f i d g e h f i

25 Emple: 9 7

26 Idetity mtri The idetity mtri Iis squre mtri with oes o the mi digol d zeros elsewhere. Emple: idetity mtri I idetity mtri I I

27 Ivertig mtri Mtri Bis the iverse of mtri Aif their produt is equl to the idetity mtri: AB BA I B A & A B If the determit of mtri is equl to zero, the the mtri is sigulr d ot ivertile.

28 Emple: Is Bthe iverse of A? A B AB 8 BA AB BA I NO Is is the iverse of C? C.. C C C C I C & C YES 7

29 System of Lier Equtios Multi-dimesiol lier equtio: K k d re slr ostts k re first-order vriles k Methods for solvig systems of lier equtios Guss-Jord elimitio geerl systems Crmer s rule simple for systems 8

30 Simulteous lier equtios equtios, ukows e writte i mtri-vetor form: K M K K Proedure:. Write the oeffiiets s mtri.. Write the vriles s olum vetor.. Write the ostts s olum vetor. 9 A s olum vetor. Mtri-vetor form M M L M O M M L L

31 Crmer s Rule for Solvig Lier Equtios system of lier equtios A Compute the followig determits:,, The solutio to the system is:, Notie how the vetor reples sigle olum i the A mtri for d.

32 Emple: Solutio:

33 system of lier equtios Compute the followig determits: A,,, The solutio to the system is:,,,,,

34 Emple: Solutio:

35 Tutorils Purplemth Your Alger Resoure