UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS
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1 UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS WTW 211 : LINEÊRE ALGEBRA / LINEAR ALGEBRA SEMESTERTOETS 1 / SEMESTER TEST 1 VAN/SURNAME: VOORNAME/FIRST NAMES: STUDENTENOMMER/STUDENT NUMBER: HANDTEKENING/SIGNATURE: LESINGGROEP/LECTURE GROUP: Maart / March 2008 TYD/TIME: 75 min PUNTE / MARKS: 30 PUNTE MARKS LEES DIE VOLGENDE INSTRUK- SIES 1. Die vraestel bestaan uit bladsye 0 tot 7 (vrae 1 tot 6). Kontroleer of jou vraestel volledig is. 2. Doen alle krapwerk 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 enige iets 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. Slegs nie-programmeerbare sakrekenaars mag gebruik word. 7. Enige navrae oor die nasienwerk moet binne drie dae nadat die toets uitgedeel is, gedoen word. Daarna word aanvaar dat alles korrek is. READ THE FOLLOWING INSTRUC- TIONS 1. The paper consists of pages 0 to 7 (questions 1 to 6). 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 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 had not been marked. 6. Only non-programmable pocket calculators may be used. 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. Outeursreg voorbehou Copyright reserved 0
2 1. Let ū 1, ū 2,..., ū m be vectors in R n. Laat ū 1, ū 2,..., ū m vektore in R n wees. (a) What is meant by w is a linear combination of ū 1, ū 2,..., ū m? [1] Wat word bedoel met w is n lineêre kombinasie van ū 1, ū 2,..., ū m? (b) Define/Definieer: span({ū 1, ū 2,..., ū m }). [1] (c) Give a precise meaning of ū 1, ū 2,..., ū m are linearly dependent. Gee die presiese betekenis van ū 1, ū 2,..., ū m is lineêr afhanklik. [1] (d) Suppose u m is a linear combination of ū 1, ū 2,..., ū m 1. Show directly from the definition of linear dependence that ū 1, ū 2,..., ū m are linearly dependent. Veronderstel u m is n lineêre kombinasie van ū 1, ū 2,..., ū m 1. Toon direk aan met behulp van die definisie van lineêr afhanklikheid dat ū 1, ū 2,..., ū m lineêr afhanklik is. [3] 1
3 2. (a) Show directly from the definition of linear dependence that the R 2 vectors [a, b] and [c, d] are linearly dependent if and only if ad = bc. Toon direk aan met behulp van die definisie van lineêr afhanklikheid dat die vektor [a, b] en [c, d] in R 2 lineêr afhanklik is as en slegs as ad = bc. [4] 2
4 (b) Find all the real numbers t for which the R 2 vectors [1 + t, 1 t] and [1 t, 1 + t] are linearly dependent. [1] Vind al die reële getalle t waarvoor die vektore [1 + t, 1 t] en [1 t, 1 + t] in R 2 lineêr afhanklik is. 3. Let { v 1, v 2,..., v m } be a linearly independent subset of R n. Laat { v 1, v 2,..., v m } n lineêr onafhanklike versameling in R n wees. (a) What is meant by { v 1, v 2,..., v m } is linearly independent? Wat beteken { v 1, v 2,..., v m } is lineêr onafhnklik? [1] (b) Show that the set { v 1, v 2,..., v k }, where k m, is linearly independent. Bewys dat die versameling { v 1, v 2,..., v m }, met k m lineêr onafhanklik is. [2] 3
5 (c) Suppose w is in R n. Show that { w, v 1, v 2,..., v m } is linearly independent if and only if span({ w, v 1, v 2,..., v m }) span({ v 1, v 2,..., v m }). Gestel w is in R n. Bewys dat { w, v 1, v 2,..., v m } lineêr afhanklik is as en slegs as span({ w, v 1, v 2,..., v m }) span({ v 1, v 2,..., v m }). [3] 4
6 4. Suppose the subset {ū, v, w} of R n is linearly independent. Determine, showing the calculations, whether or not {ū v, v + w, ū + w} is linearly independent. Gestel die versameling {ū, v, w} in R n is lineêr onafhanklik. Bepaal of {ū v, v + w, ū + w} lineêr onafhanklik is of nie (toon alle bewerkings aan). [3] 5. Let / Laat A = = [ v 1 v 2 v 3 v 4 ]. (a) Determine, showing the calculations, whether or not { v 1, v 2, v 3, v 4 } is linearly independent. Bepaal [2] 5
7 (b) Determine, showing the calculations, whether or not the vector b = is in col(a). [2] 6. (a) Define a subspace of R n. [1] 6
8 (b) Let S be the subset of R 3 consisting of all vectors [x, y, z] satisfying at least one of the two conditions x = 0 and y = 0. Determine, giving reasons, whether or not S is a subspace of R 3. [2] (c) Let T be the subset of R 3 consisting of all vectors [x, y, z] satisfying x + y + z = 0. i. Show that T is a subspace of R 3. [2] ii. Find a matrix A such that T = null(a) [1] 7
Eksterne eksaminator / External examiner: Dr. P Ntumba Interne eksaminatore / Internal examiners: Prof. I Broere, Prof. JE vd Berg, Dr.
VAN / SURNAME: UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA FAKULTEIT NATUUR- EN LANDBOUWETENSKAPPE / FACULTY OF NATURAL AND AGRICULTURAL SCIENCES DEPARTEMENT WISKUNDE EN TOEGEPASTE WISKUNDE / DEPARTMENT
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