Bu durumda ya cozum yoktur veya sonsuz cozum vardir. KIsaca cozum tek degildir. Veya cozumler birbirine lineer bagimlidir.

Transkript

1 Vektorlerin lineer bagimsiligi Ornek, Denklem Takimini Coun > - Ikinci denklemde erine ko (-) -) Sonuc: > - sartini saglaan butun ve ler her iki denklemi de coer. (, ), (, ), (, ),... Denklem takiminin sonsu coumu vardir. Bu durumda a coum oktur vea sonsu coum vardir. KIsaca coum tek degildir. Vea coumler birbirine lineer bagimlidir. Ornek, > - Ikinci denklemde erine ko (-) -) Her iki denklemi coecek ve degerleri mevcut degildir. Denklem takimi Coumsu. Ornek, Denklem Takimini Coun Denklem Takimini Coun > - Ikinci denklemde erine ko (-) Denklemin coumu tekdir. (, ) denkelmin tek coumudur. Bir lineer denklem takiminda katsailar matrisi lineer bagimli ise coum a oktur vea sonsu coum vardir. Katsailar matrisi ornek ve de.satir - ->. satir --> Birinci satiri - ile carpip. satira eklesek sifir olur. Yani birinci ve ikinci satir arasinda bir baginti vardir.

2 Vektorlerin Dogrusal Bagimsiligi Linear independence of vectors ) p[ ], p is a dimensional row vector p elemanli satir vektordur. p nin boutu dur. q, q is a dimensional column vector q elemanli sutun vektordur. q nin boutu dir Definition:,,,, n, are vectors If there are numbers a, a, a,. a n, such that a a a.a n n then vectors,,,, n are linearl dependent For two vectors, a a or a - a a - α a i.e. if the two vectors are multiples of each other then the two vectors are linearl dependent ) m, n It is clear that nm or m.n. Thus m and n are linearl dependent ) m, n There is no number α that satisfies the equation n α m. Thus m and n are linearl independent ) m, n It is clear that nm. m and n are linearl dependent ) m, n n-m. m and n are linearl dependent ) m, n There is no number α that satisfies the eqation n αm m and n are linearl independent ) m, n, p ambncp a b c a b c c c a b c b b a b c a a There is no numbers (ecept ero) that satisfies the equation ambncp. Thus the vectors m,n,p are linearl independent ) m, n, p ambncp a b c a b c c c a b c a c a a a b c a a There are no numbers (ecept a, b, c) that satisfies the equation ambncp. Thus the vectors m,n,p are linearl independent ) m, n, p a b c a b c ab a-b a b c a b a-b a b c abc replace a-b (-b) bc -bb c c The values c, a-b satisfies the equation.

3 Eample c, b, a- or c, b, a- etc satisfies the equation ambncp Vectors are linearl dependent.

4 - LINEE DENKLEMLE (Denklem saisi Bilinmeen saisi) - Tek Coum Tek Coum Sonsu Coum Sonsu Coum Coum ok Coum ok Oet Tek Coum Sonsu Coum Genisletilmis matris Gaus eliminason ontemile alt ucgeni sifirlamaa calis. Eger en son satir tamamen sifir ise coum sonsu tane Eger en son satirin en son elemani sifirdan farkli ise tek coum Eger en son satirin sadece en son elemani sifirdan farkli ise tek coum ok Coum Yok

5 Multiple Solution No Solution w w w -w - Unique Solution w

6 A: katsailar matrisi ~ A : Genisletilmis matris. r : bilinmeen saisi. (nn icin denklem saisi) rank Arank A ~ r >tek coum rank Arank A ~ < r >sonsu coum rank A< rank A ~ > coum ok A ank Arank A ~ r Tek Coum A - ank A, rank ~ A Coum ok A rank Arank ~ A < Sonsu coum Find the solutions for the equations - w, -w The Augmented matri is - - The echelon form is - - n, rank Arank A ~ < multiple solution n-r- variables are arbitraril selected. ( free variables) - w, w- First solution Set w and solve for and : -, - -, V[ w][ - ] Second solution Set w and solve for and : -, - -, V[ w][ - ] Third Solution Set, and solve for and : - w, w- w-, w, V[ w][ -] Forth Solution Set, - w, w- w-; w; V[ -] Find the solution space for the homogenous sstem since rank An the sstem has empt solution space. (no solution eists ecept the trivial solution ) Find the solution space for the homogenous sstem rank Ar n; n-r- The sstem has one nontrivial solution. To find that solution set ; and use equations - ; ; Solution set is V[ ][ - ] There is onl one (n-r) independent solution. All other solutions will be multiples of this single solution. For eample set ; V[ ][ - ] It is clear that V*V. i.e V and V are linearl dependent. Find the solution space for the homogenous sstem A rank Ar n; n-- variables are freel selected. Set, calculate. (-). V[- ]. Set, - V[- ] All other solutions will be linearl dependent.

7 Gaus Eliminason Yontemile Lineer denklemlerin coumu Augmented Matri Birinci satiri ile carp ikinci satira ilave et. * * * * Birinci satiri -. ile carp ucuncu satira ilave et...*.*.*.* Ikinci satiri - ile carp ikinci satira ilave et. * ) * ( * Sonuc denklemleri a. Sondan Baslaarak denklemleri co.. *.. * *. During the process we converted the coefficient matri into upper triangular Form

8 Aciklama ( E) ( E) ( E) Birinci esitligin (E) iki tarafini - ile carp (E) Elde edilen bu eni esitligi ikinci esitlige (E) ekle. ( E) Multipl both side of equation (E) b.. Add this modified equation (E) to equation (E). ( E) Now Solve (E) and (E) ( E) ( E) Multipl equation (E) b and add to equation (E) Thus. Substitute into equation (E) and obtain. *. Substitute and. into the first equation of (E) and solve for. *. *.. esult -..

9 Gaus Jordan (diagonal kosegen form) Yontemile Denklem Coumu Hedef ,.,