Eksterne eksaminator / External examiner: Dr. P Ntumba Interne eksaminatore / Internal examiners: Prof. I Broere, Prof. JE vd Berg, Dr.
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1 VAN / SURNAME: UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA FAKULTEIT NATUUR- EN LANDBOUWETENSKAPPE / FACULTY OF NATURAL AND AGRICULTURAL SCIENCES DEPARTEMENT WISKUNDE EN TOEGEPASTE WISKUNDE / DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS VOORNAME / FIRST NAMES: WTW LINEÊRE ALGEBRA / LINEAR ALGEBRA SEMESTERTOETS 1 / SEMESTER TEST 1 19 Maart 2013 om 17:30 / 19 March 2013 at 17:30 TYD / TIME: 60 min PUNTE / MARKS: 32 STUDENTENOMMER / STUDENT NUMBER: HANDTEKENING / SIGNATURE: SEL NR / CELL NO: Eksterne eksaminator / External examiner: Dr. P Ntumba Interne eksaminatore / Internal examiners: Prof. I Broere, Prof. JE vd Berg, Dr. JH vd Walt PUNTE / MARKS Q1 Q2 Q3 Q4 TOTAAL TOTAL MAKS / MAX Merk u lesinggroep AFRIKAANS ENGLISH Tick your lecture group INSTRUKSIES 1. Die vraestel bestaan uit hierdie voorblad en nog vier bladsye wat vier vrae bevat. Kontroleer of jou vraestel volledig is. 2. Die gebruik van alle elektroniese toerusting is verbode: Geen kandidaat mag enige i-pad, selfoon, sakrekenaar, ens. gebruik tydens die skryf van hierdie vraestel nie. 3. Doen alle rofwerk op die teenblad. Dit word nie nagesien nie. 4. As jy meer as die beskikbare ruimte vir n antwoord nodig het, gebruik dan ook die teenblad en dui dit asseblief duidelik aan. 5. Geen potloodwerk of enige iets wat in rooi ink gedoen is, word nagesien nie. 6. As jy korrigeerink (Tipp-Ex of soortgelyk) gebruik, verbeur jy die reg om werk wat nagesien is te bevraagteken of te beweer dat werk nie nagesien is nie. 7. Enige navrae oor die nasienwerk moet binne drie dae nadat die toets uitgedeel is, gedoen word. Daarna word aanvaar dat alles korrek is. Outeursreg voorbehou INSTRUCTIONS 1. The paper consists of this cover page and four more pages containing four questions. Check whether your paper is complete. 2. The use of all electronic equipment is forbidden: No candidate is allowed to use any i-pad, cell phone, calculator, etc. while writing this paper. 3. Do all scribbling on the facing page. It will not be marked. 4. If you need more than the available space for an answer, use the facing page and please indicate it clearly. 5. No pencil work or any work in red ink will be marked. 6. If you use correcting fluid (Tipp-Ex or similar), you lose the right to question the marking or claim that work has not been marked. 7. Any queries about the marking must be done within three days after the tests have been handed out. After that we assume that everything is in order. Copyright reserved
2 1 1. Let S = {v 1, v 2,..., v k } be a subset of R m. / Laat S = {v 1, v 2,..., v k } n deelversameling van R m wees. (a) Define the following: The span of S. / Definieer die volgende: Die span van S. [2] (b) If v and w are in the span of S and a and b are scalars, show that av + bw is in the span of S. / As v en w in die span van S is en a en b is skalare, bewys dat av + bw in die span van S is. [2] (c) Is the zero vector 0 in the span of every set of vectors? Motivate your answer. / Is die nulvektor 0 in die span van elke versameling vektore? Motiveer jou antwoord. [2] (d) Is the vector [1 2 3] in the span of the set of vectors {[1 0 1], [2 2 1]}? Motivate your answer. / Is die vektor [1 2 3] in die span van die versameling vektore {[1 0 1], [2 2 1]}? Motiveer jou antwoord. [3]
3 2 2. Let S = {u 1, u 2,..., u m } be a subset of R n. / Laat S = {u 1, u 2,..., u m } n deelversameling van R n wees. (a) Define the following: S is linearly dependent. / Definieer die volgende: S is lineêr afhanklik. [2] (b) Now define the concept S is linearly independent. / Definieer nou ook die begrip S is lineêr onafhanklik. [1] (c) Prove the following result: If S = {u 1, u 2,..., u m } is linearly dependent, then at least one of the vectors of S can be expressed as a linear combination of the others. / Bewys die volgende resultaat: As S = {u 1, u 2,..., u m } lineêr afhanklik is, dan kan minstens een van die vektore in S geskryf word as n lineêre kombinasie van die ander. [3] (d) If the zero vector 0 is one of the vectors in the set S = {u 1, u 2,..., u m }, can we then say that S is necessarily linearly dependent? Prove your claim in detail. / As die nulvektor 0 een van die vektore in die versameling vektore S = {u 1, u 2,..., u m } is, kan ons dan sê dat S noodwendig lineêr afhanklik is? Bewys jou bewering noukeurig. [3]
4 3 3. (a) Using any systematic method, show that the trivial solution x = y = z = 0 is the only solution for the following system of equations: / Gebruik enige sistematiese metode om te bewys dat die triviale oplossing x = y = z = 0 die enigste oplossing vir die volgende stelsel vergelykings is: x y + z = 0; 2x + y = 0; y 4z = 0. [3] (b) Now consider the coefficient matrix of the system of equations in (a): it is A = Beskou nou die koëffisiëntematriks van die stelsel vergelykings in (a): dit is Without any further calculations, say whether the set S of column vectors of A is a linearly independent or a linearly dependent set of vectors. Motivate your answer. / Sonder enige verdere berekeninge, sê of die versameling S van kolomvektore van A n lineêr onafhanklike versameling of n lineêr afhanklike versameling vektore is. Motiveer jou antwoord. [2]. (c) With S the set of column vectors of A as above, is there a vector v in R 3 such that S {v} is necessarily a linearly independent set of vectors? If so, give an example of such a vector, if not, say so; whatever your answer, motivate it. / Met S die versameling kolomvektore van A soos hierbo, is daar n vektor v in R 3 só dat S {v} n lineêr onafhanklike versameling vektore is? Indien wel, gee n voorbeeld van só n vektor, indien nie, sê so; wat jou antwoord ookal is, motiveer dit. [3]
5 4 4. Let S = {v 1, v 2,..., v m } and T = {v 1, v 2,..., v n } with n m (so that S T ). Prove the following: / Laat S = {v 1, v 2,..., v m } en T = {v 1, v 2,..., v n } met n m (sodat S T ). Bewys die volgende: (a) If S is linearly dependent, then T is also linearly dependent. / As S lineêr afhanklik is, dan is T ook lineêr afhanklik. [3] (b) span(s) span(t ). [3] The end
UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS
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