Section 7.3, Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors (the variable vector of the system) and

Transcription

1 Sec. 7., Boyce & DiPrim, p. Sectio 7., Systems of Lier Algeric Equtios; Lier Idepedece, Eigevlues, Eigevectors I. Systems of Lier Algeric Equtios.. We c represet the system usig mtrices d vectors s follows: Let A = (the coefficiet mtri of the system) = (the vrile vector of the system) d = (the ohomogeeous vector of the system). The the system c e represeted s A =. If, the system is sid to e ohomogeeous. If = the system is sid to e homogeeous. The mtri (A ) = is referred to s ugmeted mtri for the system A =. Emple. Idetify the mtri A d vectors d d set up the ugmeted mtri for the followig system

2 II. Solvig system of lgeric equtios A. Solvig systems usig the iverse mtri. If A =, where A is osigulr squre mtri, the A - A = I A - A = A - = A -. Emple. Solve the system usig A -. B. Solvig systems usig row opertios o the ugmeted mtri. A ugmeted mtri is trsformed ito row-equivlet mtri y formig y of the followig row opertios:. Two rows re iterchged (R i R j ).. A row is multiplied y ozero costt (kr i R i ).. A costt multiple of oe row is dded to other row (kr j + R i R i ). [Note: The rrow mes replces. ] Emple. Perform the followig opertios o the ugmeted mtri of. 6 8 (Ech time egi with the origil ugmeted mtri.) ) R R ) R R c) R + R R Reduced Mtri A mtri is reduced mtri or is sid to e i reduced form if:. Ech row cosistig etirely of zeros is elow y row hvig t lest oe ozero elemet.. The leftmost ozero elemet i ech row is.. All other elemets i the colum cotiig the leftmost of give row re zeros.. The leftmost i y row is to the right of the leftmost i the row ove. Sec. 7., Boyce & DiPrim, p.

3 Sec. 7., Boyce & DiPrim, p. Emple. If the mtrices elow re ot i reduced form, idicte which coditio(s) is/re violted for ech mtri. Use row opertios to fully reduce the mtri. ) 5 ) c) d) Emple 5. Write the lier system correspodig to ech reduced mtri d solve. )

4 Sec. 7., Boyce & DiPrim, p. ) 5 Emple 6. Solve Emple 7. Fid the ifiitely my solutios of the system with mtri. Emple 8. Fid the iverse of A =.

5 III. Lier idepedece of vectors. Defiitio: The vectors (), (),..., (k) re lierly idepedet if wheever c () + c () + + c k (k) = (zero vector) ll of the c i must e zero. If vectors re ot lierly idepedet they re lierly depedet. Note tht if vector is multiple of other vector, the two vectors re depedet. () () () Emple 7. Determie if the vectors,, re lierly idepedet or depedet y usig the defiitio. If they re depedet determie the reltioship of depedecy. Emple 8. Are the eigevectors () = depedet? d () = lierly idepedet or Sec. 7., Boyce & DiPrim, p. 5

6 IV. Eigevlues d eigevectors Suppose A is mtri. The ozero vector is eigevector of the mtri A if A = for some (possily comple) eigevlue. Geometriclly, multiplictio of eigevector y A does ot chge the directio of ut oly its oriettio or legth. Emple 9.. Give A =. Show tht () = d () = re eigevectors of A d determie the correspodig eigevlues.. Show tht if is eigevector of A correspodig to, the k is lso eigevector of A correspodig to. Note tht this implies tht for every eigevlue there re ifiitely my eigevectors - ll of them multiples of oe eigevector. How to fid eigevlues d eigevectors.. To fid eigevlues, solve the chrcteristic equtio, det(a I) =. Emple. Use det(a I) = to fid the eigevlues of A =. Sec. 7., Boyce & DiPrim, p. 6

7 . To fid the correspodig eigevectors, solve (A I) =. Note tht ecuse there re ifiitely my eigevectors correspodig to y eigevlue, there will e t lest oe row of zeros i your reduced mtri (A ). Emple. Fid the eigevectors of A =. Sec. 7., Boyce & DiPrim, p. 7