Supplement: Gauss-Jordan Reduction

Transcription

1 Suppleme: Guss-Jord Reducio. Coefficie mri d ugmeed mri: The coefficie mri derived from sysem of lier equios m m m m is m m m A O d he ugmeed mri derived from he ove sysem of lier equios is [ ] m m m m A O where m. Emple: For sysem of lier equios

2 he coefficie mri is d he ugmeed mri is.. oivio of Guss-Jord reducio: oivig Emple: Suppose we hve he followig sysem of lier equios d he ssocied ugmeed mri () () - () ) ( ) ( () () - () ) ( () ) ( () () () ) *( () ) ( () () ()

3 () ( () () ) () *( ) () ( () () ) () () () () () () () () ( - () () ) () *( ) () ( - () () ) () *( ) I he ove moivig emple here re elemery operios o he lier equios h c e used o oi he soluio. They re. Ierchge wo equios.. uliply equio y ozero cos.. Add muliple of equio o oher equio. I is simpler o use he ugmeed mri o represe he operios o hese equios sice we do eed o keep wriig he vrile. I ddiio he procedure sed o he ugmeed mri c e implemeed wih compuer sofwre. There re elemery row operios of he ugmeed mri correspodig o elemery operios o he lier equios. They re. Ierchge wo rows.. uliply row y ozero cos.. Add muliple of row o oher row.

4 Noe h he ls ugmeed mri hs he followig properies: For ech row he firs ozero ery is (clled ledig ). For wo successive rows he ledig i he higher row ppers frher o he lef h he ledig i he lower row. If colum cois ledig he ll oher eries i h colum re. Coclusio: i migh e more efficie o solve he sysem of lier equios sed o he ugmeed mri vi few elemery row operios. Also if he fil ugmeed mri is similr o he oe i he moivig emple he soluio migh e esy o oi. Noe: he procedure i he moivig emple is i fc Guss-Jord reducio.. Reduced row echelo form d elemery row operios: I ove moivig emple he key o solve sysem of lier equios is o rsform he origil ugmeed mri o some mri wih some properies vi few elemery row operios. As mer of fc we c solve y sysem of lier equios y rsformig he ssocie ugmeed mri o mri i some form. The form is referred o s he reduced row echelo form. Defiiio of mri i reduced row echelo form: A mri i reduced row echelo form hs he followig properies:. All rows cosisig eirely of re he oom of he mri.. For ech ozero row he firs ery is. The firs ery is clled ledig.. For wo successive ozero rows he ledig i he higher row ppers frher o he lef h he ledig i he lower row.. If colum cois ledig he ll oher eries i h colum re. Noe: mri is i row echelo form s he mri hs he firs properies.

5 Emple: d re he mrices i reduced row echelo form. The mri is o i reduced row echelo form u i row echelo form sice he mri hs he firs properies d ll he oher eries ove he ledig i he hird colum re o. The mri re o i row echelo form (lso o i reduced row echelo form) sice he ledig i he secod row is o i he lef of he ledig i he hird row d ll he oher eries ove he ledig i he hird colum re o. Defiiio of elemery row operio: There re elemery row operios:. Ierchge wo rows. uliply row y some ozero cos. Add muliple of row o oher row.

6 Emple: A. Ierchge rows d of A uliply he hird row of A y uliply he secod row of A y - he dd o he hird row of A Impor resul: Every ozero m mri c e rsformed o uique mri i reduced row echelo form vi elemery row operios. If he ugmeed mri [ ] A c e rsformed o he mri i reduced row echelo form [ ] d C vi elemery row operios he he soluios for he lier sysem correspodig o [ ] d C is ecly he sme s he oe correspodig o [ ] A. Emple: () () () ()

7 ) ( () () () () () () () () () () () / ) *( () () () () () () / ) ( () () () () () / ) ( ()

8 8 () () () - () / / / / ) ( () () () () () - () / / / / / ) *( () () () () () - () / / / / / () () () () () - () / / / / / /

9 ) *( () () () () () - () / / / / / / ) *( () () () () () - () / / / / / ) *( () () () () () - () / / / / Therefore

10 c e rsformed o he uique mri i reduce row echelo form / / / / vi elemery row operios. The lier sysem () () () () hs he ecly he sme soluio s he lier sysem () () () - () The soluio for he lier sysem correspodig o he ugmeed mri i reduced row echelo form is R

11 / / / / ) ( / ) / ( ) / ( ) ( / The ove soluios re lso he soluios for he origil lier sysem.. Guss-Jord reducio: Sep : Form he ugmeed mri correspodig o he sysem of lier equios. Sep : Trsform he ugmeed mri o he mri i reduced row echelo form vi elemery row operios. Sep : Solve he lier sysem correspodig o he mri i reduced row echelo form. The soluio(s) re lso for he sysem of lier equios i sep. Emple: Solve for he followig lier sysem: - - [soluio:] The Guss-Jord reducio is s follows: Sep : The ugmeed mri is. Sep : Afer elemery row operios he mri i reduced row echelo form is

12 . Sep : The lier sysem correspodig o he mri i reduced row echelo form is The soluios re R Numer of soluios of sysem of lier equios: For y sysem of lier equios precisely oe of he followig is rue. I. The sysem hs ecly oe soluio. II. The sysem hs ifiie umer of soluios. III. The sysem hs o soluio. Noe: he lier sysem wih les oe soluio is clled cosise d he lier sysem wih o soluio is clled icosise. Emple: I. Ecly oe soluio: Solve for he followig sysem: 8

13 [soluio:] The Guss-Jord reducio is s follows: Sep : The ugmeed mri is 8. Sep : The mri i reduced row echelo form is Sep : The soluio is II. Ifiie umer of soluios: Solve for he followig sysem: [soluio:] The Guss-Jord reducio is s follows: Sep : The ugmeed mri is Sep : The mri i reduced row echelo form is Sep :

14 The lier sysem correspodig o he mri i reduced row echelo form is The soluios re R III. No soluio: Solve for he followig sysem: [soluio:] The Guss-Jord reducio is s follows: Sep : The ugmeed mri is Sep : The mri i reduced row echelo form is Sep : The lier sysem correspodig o he mri i reduced row echelo form is

15 Sice here is o soluio.. Homogeeous sysems: A lier sysem of he form m m m is clled homogeeous sysem. Noe: A homogeeous sysem mus hve les oe soluio. Th is every homogeeous sysem of lier equios is cosise. Noe: The soluio is clled he rivil soluio o he homogeeous sysem. A soluio K is clled orivil soluio if o ll he i re. Impor resul: A homogeeous sysem of m equios i ukow vriles lwys hs ifiie umer of soluios if m <. e p e soluio of he ohomogeeous sysem correspodig o he ugmeed mri [ A ] d le h e he soluio of he ssocied homogeeous sysem [ A ]. The p h is lso soluio of he ohomogeeous sysem. Every soluio of he ohomogeeous sysem correspodig o he ugmeed mri [ ] A c e wrie s p h where p e

16 soluio of he ohomogeeous sysem d h e he soluio of he ssocied homogeeous sysem [ ] A. Emple: For he ohomogeeous lier sysem he lier sysem correspodig o he mri i reduced row echelo form is The soluios re R The p d R h. Noe h R h re he soluios for he homogeeous lier sysem