EXAMINATION / EKSAMEN 19 JUNE/JUNIE 2013 AT / OM 08:00

Transcription

1 UNIVERSITY OF PRETORIA / UNIVERSITEIT VAN PRETORIA FACULTY OF NATURAL AND AGRICULTURAL SCIENCES / FAKULTEIT NATUUR- EN LANDBOUWETENSKAPPE DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS / DEPARTEMENT WISKUNDE EN TOEGEPASTE WISKUNDE WTW LINEAR ALGEBRA / LINEÊRE ALGEBRA EXAMINATION / EKSAMEN 19 JUNE/JUNIE 2013 AT / OM 08:00 TIME/TYD: 120 min. MARKS/PUNTE: 60 SURNAME/VAN: FIRST NAMES/VOORNAME: STUDENT NUMBER/STUDENTENOMMER: SIGNATURE/HANDTEKENING: CELL NUMBER/SELNOMMER: Internal examiners / Interne eksaminatore: Prof. I. Broere, Prof. J.E. van den Berg, Dr. J.H. van der Walt External examiner / Eksterne eksaminator: Dr. P. Ntumba PUNTE MARKS Q1 Q2 Q3 Q4 Q5 Q6 TOTAL READ THE FOLLOWING INSTRUC- TIONS 1. The paper consists of pages 0 to 7. Check whether your paper is complete. 2. Do all scribbling on the facing page. It will not be marked. 3. If you need more than the available space for an answer, use the facing page and please indicate it clearly. 4. No pencil work, or any work done in red ink, will be marked. 5. If you use correcting fluid ( Tipp-Ex ), you lose the right to question the marking or to indicate work that has not been marked. 6. Any queries about the marking must be done during the perusal periods of the scripts. The date for perusals will be announced on ClickUP. 7. No candidate is allowed to use any i-pad, cell phone, calculator, etc. while writing this paper. LEES DIE VOLGENDE INSTRUK- SIES 1. Die vraestel bestaan uit bladsye 0 tot 7. Kontroleer of jou vraestel volledig is. 2. Doen alle rofwerk op die teenblad. Dit word nie nagesien nie. 3. As jy meer as die beskikbare ruimte vir n antwoord nodig het, gebruik dan ook die teenblad en dui dit asseblief duidelik aan. 4. Geen potloodwerk, of enigiets wat in rooi ink gedoen is, word nagesien nie. 5. As jy korrigeerink ( Tipp-Ex ) gebruik, verbeur jy die reg om te kla oor werk wat nie nagesien is nie of wat verkeerd nagesien is. 6. Enige navrae oor die nasienwerk moet gedoen word tydens die besigtigingperiode van die vraestelle. Die datum van hierdie periode sal op ClickUP aangekondig word. 7. Geen kandidaat mag enige i-pad, selfoon, sakrekenaar, ens. gebruik tydens die skryf van hierdie vraestel nie. Copyright reserved Outeursreg voorbehou 0

2 1. Let / Laat A = , x = x y z and / en b = (a) Find all the solutions to the homogeneous system of equations Ax = 0. / Vind al die oplossings van die homogene stelsel vergelykings Ax = 0. [4] (b) What is the dimension of the null space of the matrix A? Motivate your answer. / Wat is die dimensie van die nulruimte van die matriks A? Motiveer jou antwoord. [1] (c) What is the rank of the matrix A? Motivate your answer. / Wat is die rang van die matriks A? Motiveer jou antwoord. [2] (d) Describe the column space of the matrix A as a set of vectors or by finding a basis for it. / Beskryf die kolomruimte van die matriks A as n versameling vektore of deur n basis daarvoor te vind. [3] (e) How many solutions are there for the system of equations Ax = b? Motivate your answer. / Hoeveel oplossings is daar vir die stelsel vergelykings Ax = b? Motiveer jou antwoord. [2] 1

3 2. (a) Prove the following theorem: Let A be a matrix whose entries are real numbers. For the system of equations Ax = b, exactly one of the following is true: a. There is no solution; b. There is a unique solution and c. There are infinitely many solutions. / Bewys die volgende stelling: Laat A n matriks wees waarvan die inskrywings reële getalle is. Vir die stelsel vergelykings Ax = b is presies een van die volgende waar: a. Daar is geen oplossing; b. Daar is n unieke oplossing; en c. Daar is oneindig veel oplossings. [5] (b) Complete the following definition: A subspace of R n is.... / Voltooi die volgende definisie: n Deelruimte van R n is..... [2] (c) Prove the following theorem: Let A be an m n matrix and let N be the set of solutions of the homogeneous linear system Ax = 0. Then N is a subspace of R n. / Bewys die volgende stelling: Laat A n m n matriks wees en laat N die versameling oplossings van die homogene stelsel Ax = 0 wees. Dan is N n deelruimte van R n. [3] 2

4 3. (a) Let T : R n R m be a function. Define the following concept: T is a linear transformation. / Laat T : R n R m n funksie wees. Definieer die volgende begrip: T is n lineêre transformasie. [2] (b) Let T : R n R m be a linear transformation and let {e 1, e 2,..., e n } be the standard basis of R n. Describe the standard matrix [T ] of T. / Laat T : R n R m n lineêre transformasie wees en laat {e 1, e 2,..., e n } die standaard basis van R n wees. Beskryf die standaard matriks [T ] van T. [1] (c) Prove the following result: Let T : R n R n be an invertible linear transformation. Then the standard matrix [T ] is an invertible matrix and [T 1 ] = [T ] 1. / Bewys die volgende: Laat T : R n R n n inverteerbare lineêre transformasie wees. Dan is die standaardmatriks [T ] n inverteerbare matriks en [T 1 ] = [T ] 1. [4] 3

5 4. Let / Laat A = (a) Write down the characteristic polynomial and the eigenvalues of the matrix A. / Skryf die karakteristieke veelterm en die eiewaardes van die matriks A neer. [3] (b) Describe the eigenspace of the smallest eigenvalue (1) of the matrix A; you may describe it as a set of vectors or by finding a basis for it. / Beskryf die eieruimte van die kleinste eiewaarde (1) van A; jy mag dit as n versameling vektore beskryf of dit beskryf deur n basis daarvoor te vind. [3] (c) Write down the dimension of the eigenspace you described in Question 4(b) above. / Skryf die dimensie van die eieruimte wat jy in Vraag 4(b) hierbo beskryf het neer. [1] (d) The algebraic multiplicity of each eigenvalue of A can easily seen to be 1. What is the geometric multiplicity of each eigenvalue of A? Motivate your answer stating clearly any theorems used in your argument. / Die algebraïese multiplisiteit van elke eiewaarde van A is duidelik 1. Wat is die meetkundige multiplisiteit van elke eiewaarde van A? Motiveer jou antwoord deur n geskikte argument waarin jy elke resultaat wat jy gebruik formuleer. [3] 4

6 5. (a) Prove the following implication (which is part of the Diagonalization Theorem): If A is an n n matrix and the union of bases of the eigenspaces of A contains n vectors, then the algebraic multiplicity of each eigenvalue of A equals its geometric multiplicity. / Bewys die volgende implikasie (wat deel is van die Diagonaliseringstelling): As A n n n matriks is en en die vereniging van basisse van die eieruimtes van A bevat n vektore, dan is die algebraïese multiplisiteit van elke eiewaarde van A gelyk aan die meetkundige multiplisiteit daarvan. [4] (b) Prove the following: If A and B are n n matrices and there exists an invertible matrix P such that A = P 1 BP, then there also exists for every positive integer m an invertible matrix Q such that A m = Q 1 B m Q. / Bewys die volgende: As A en B beide n n matrikse is en daar bestaan n inverteerbare matriks P só dat A = P 1 BP, dan bestaan daar ook vir elke positiewe heelgetal m n inverteerbare matriks Q só dat A m = Q 1 B m Q. [4] (c) Use the result from Question 5(b) above to prove the following result: If A is a diagonalizable matrix, then A m is also diagonalizable for every positive integer m. / Gebruik die resultaat van Vraag 5(b) hierbo om die volgende resultaat te bewys: As A n diagonaliseerbare matriks is, dan is A m ook diagonaliseerbaar vir elke positiewe heelgetal m. [2] 5

7 [ (a) i. Let / Laat A = 2 3 ]. There exists an invertible matrix P and a diagonal matrix D such that P 1 AP = D; find such matrices P and D. / Daar bestaan n inverteerbare matriks P en n diagonaalmatriks D só dat P 1 AP = D; vind sulke matrikse P en D. [4] ii. Use the result of Question 6(a)i to calculate A 5. / Gebruik die resultaat van Vraag 6(a)i om A 5 te bereken. [3] 6

8 (b) Prove the following: If B and C are n n matrices, each with n distinct eigenvalues but with the same eigenvectors, then BC = CB. / Bewys die volgende: As B en C beide n n matrikse is, elk met n verskillende eiewaardes maar met dieselfde eievektore, dan is BC = CB. [4] 7