EXAMINATION / EKSAMEN 17 JUNE/JUNIE 2011 AT / OM 12:00 Q1 Q2 Q3 Q4 Q5 Q6 TOTAL

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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 2 - LINEAR ALGEBRA / LINEÊRE ALGEBRA EXAMINATION / EKSAMEN 7 JUNE/JUNIE 20 AT / OM 2:00 TIME/TYD: 20 min. MARKS/PUNTE: 5 SURNAME/VAN: FIRST NAMES/VOORNAME: STUDENT NUMBER/STUDENTENOMMER: SIGNATURE/HANDTEKENING: CELL NUMBER/SELNOMMER: Internal examiners / Interne eksaminatore: Prof. I. Broere, Dr. S. Mutangadura External examiner / Eksterne eksaminator: Dr. P. Ntumba PUNTE MARKS Q Q2 Q3 Q4 Q5 Q6 TOTAL READ THE FOLLOWING INSTRUC- TIONS LEES DIE VOLGENDE INSTRUK- SIES. The paper consists of pages 0 to 7. Check. Die vraestel bestaan uit bladsye 0 tot 7. Kontroleer whether your paper is complete. of jou vraestel volledig is. 2. Do all scribbling on the facing page. It will not be marked. 2. Doen alle rofwerk op die teenblad. Dit word nie nagesien nie. 3. If you need more than the available space for an answer, use the facing page and please indicate it clearly. 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. No pencil work, or any work done in red ink, will be marked. 4. Geen potloodwerk, of enigiets wat in rooi ink gedoen is, word nagesien nie. 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. 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. Any queries about the marking must be done 6. Navrae oor die nasienwerk kan slegs gedoen word during the perusal period. The date for perusals tydens die besigtigingperiode; die datum hiervoor will be announced on ClickUP. sal op ClickUP aangekondig word. 7. Non-programmable calculators may be used. 7. Nie-programmeerbare sakrekenaars mag gebruik word. Copyright reserved Outeursreg voorbehou 0

2 . (a) Let S = {v, v 2,..., v k } be a set of vectors in R n. Define the span of S. / Laat S = {v, v 2,..., v k } n versameling vektore in R n wees. Definieer die span van S. [] (b) Prove that / Bewys dat span 0 0, 0, = R 3. [2] (c) Let S and T be finite nonempty subsets of R n with S T. Prove that span(s) span(t ). / Laat S en T eindige nie-le e deelversamelings van R n wees met S T. Bewys dat span(s) span(t ). [2]

3 2. Let A be an m n matrix. / Laat A n m n matriks wees. (a) Prove the following: If x 0 is in the null space of A and c is any scalar, then cx 0 is also in the null space of A. / Bewys die volgende: As x 0 in die nulruimte van A is en c is enige skalaar, dan is cx 0 ook in die nulruimte van A. [2] (b) Now prove the following theorem: If A is a matrix with real entries, then exactly one of the following is true for every system Ax = b of linear equations: (i). There is no solution. (ii). There is a unique solution. (iii). There are infinitely many solutions. / Bewys nou die volgende stelling: As A n matriks met re ele inskrywings is, dan is presies een van die volgende waar vir elke stelsel lineêre vergelykings Ax = b: (i). Daar is geen oplossing. (ii). Daar is n unieke oplossing. (iii). Daar is oneindig veel oplossings. [4] 2

4 [ ] [ ] [ ] (c) and are solutions of the system of equations 2x 3y = 7; 4x 4 = 6y. / en [ ] 2 is oplossings vir die stelsel vergelykings 2x 3y = 7; 4x 4 = 6y. (i) How many solutions does this system have? Motivate your answer. / Hoeveel oplossings het hierdie stelsel? Motiveer jou antwoord. [2] (ii) Is the set of all solutions of this system a subspace of R 2? Motivate your answer. / Is die versameling van al die oplossings van hierdie stelsel n deelruimte van R 2? Motiveer jou antwoord. [2] 3. Let / Laat A = (a) Describe the row space row(a) of A by finding a basis for it. / Beskryf die ryruimte row(a) van A deur n basis daarvoor te bereken. [3] (b) Calculate rank(a). / Bereken rank(a). [] (c) Formulate a theorem which links the nullity of a matrix to the concepts handled in (a) and (b) above and use it to calculate nullity(a). / Formuleer n stelling wat die nulheidsgraad van n matriks verbind met die begrippe wat in (a) en (b) ter sprake is en gebruik dit om nullity(a) te bereken. [2] 3

5 4. Let T : R n R m be a function. / Laat T : R n R m n funksie wees. (a) Prove the following result: If there is an m n matrix A such that T (x) = Ax for every x R n, then T is a linear transformation. / Bewys die volgende resultaat: As daar n m n matriks A is só dat T (x) = Ax vir elke x R n, dan is T n lineêre transformasie. [3] (b) Remember that the standard matrix of [ the linear transformation ] S m (of R 2 ) consisting of the projection onto the line y = mx is [S m ] = m +m 2 +m 2 m m 2 +m 2 +m 2. Use this result to derive the standard matrix of the linear transformation T m (of R 2 ) consisting of the reflection about the line y = mx. / Onthou dat die standaard matriks van die lineêre [ transformasie ] S m (van R 2 ) bestaande uit die projeksie na die lyn y = mx gelyk aan [S m ] = m +m 2 +m 2 m m 2 +m 2 +m 2 is. Gebruik hierdie resultaat om die standaard matriks van die lineêre transformasie T m (van R 2 ) bestaande uit die refleksie om die lyn y = mx af te lei. [4] (c) For which value(s) of m is the (projection) linear transformation S m a - function? Motivate your answer. / Vir welke waarde(s) van m is die (projeksie) lineêre transformasie S m n - funksie? Motiveer jou antwoord. [2] 4

6 5. Let A be an n n matrix. / Laat A n n n matriks wees. (a) Define an eigenvalue and an eigenvector of A. / Definieer n eiewaarde en n eievektor van A. [2] (b) Define an eigenspace of A. / Definieer n eieruimte van A. [2] [ ] [ 2 (c) Let A = and x = 0 4 [ ] [ 2 A = en x = 0 4 ]. Use the theory of eigenvalues to calculate A 3 x. / Laat ]. Gebruik die teorie van eiewaardes om A 3 x te bereken. [6] 5

7 6. Let A and B be n n matrices. / Laat A en B n n matrikse wees. (a) Define the following concept: A is similar to B. / Definieer die volgende begrip: A is gelyksoortig aan B. [2] (b) Now prove the following result: If A is similar to B, then A and B have the same characteristic polynomial. / Bewys nou die volgende resultaat: As A gelyksoortig is aan B, dan het A en B dieselfde karakteristieke veelterm. [3] (c) Suppose A has n linearly independent eigenvectors p, p 2,..., p n with corresponding eigenvalues λ, λ 2,..., λ n. Without giving details of the proof, show how a matrix P and a diagonal matrix D can be constructed so that P AP = D. / Veronderstel A het n lineêr onafhanklike eievektore p, p 2,..., p n met bybehorende eiewaardes λ, λ 2,..., λ n. Sonder om besonderhede van die bewys te gee, wys hoe n matriks P en n diagonaalmatriks D gekonstrueer kan word só dat P AP = D. [3] 6

8 (d) Let / Laat C = If possible, find a matrix P that diagonalises C. / Vind, indien moontlik, n matriks P wat C diagonaliseer. [3] The end / Die einde Enjoy your winter break / Geniet die winterreses! 7