MAT3701 LINEAR ALGEBRA III January/February 2021

UNIVERSITY EXAMINATIONS UNIVERSITEITSEKSAMENS




MAT3701 January/February 2021

LINEAR ALGEBRA III
Duration : 2 Hours 100 Marks

EXAMINERS :
FIRST : PROF JD BOTHA EXTERNAL : PROF LM PRETORIUS



This is an open book examination. The examination question paper remains the property of the University of South Africa.


This examination question paper consists of 3 pages.

The exam is IRIS invigilated. This requires that IRIS be activated at the start of the exam and kept on for the full duration of the
exam. The originality of submissions will also be verified with Turnitin.

Answer all the questions and show all calculations. Since this is an open book examination, if the solution to a similar question is
given, no marks will be awarded.

Please stop writing at the end of the official exam period.You then have one hour in which to scan your answer sheets (please
number the pages clearly and write your student number and module code on the front page) and submit it as a pdf file under the
name studentnumber MAT3701 on this, the myExams platform. No submissions will be accepted after the additional one
hour uploading period.




TO UPLOAD YOUR ANSWER FILE
1. When you have finished writing and created a pdf answer file, go to Submission on the Exam site and under Attachments,
click the Choose file button next to Select a file from computer. Then browse your computer for your answer file and
select it for uploading. Remember to submit your answer file as a pdf file under the name studentnumber MAT3701.
2. Once you have attached your answer file, the name of the file as well as the file size and upload time stamp will bedisplayed
under Attachments.
3. Tick the “Honour Pledge” button if you agree.
4. Click the “Submit” button.
5. After you have finished uploading your answer file, click on the “Submit” button on the IRIS pop-up screen. IRIS willthen upload
your session recording files. Remember not to close the window until IRIS is finished.
2 MAT3701
January/February 2021


QUESTION 1

Let T : V → V be a linear operator on a finite-dimensional vector space V over C such that T2 = I.

(1.1) Show that R(T + I) ⊆ N(T − I) and R(T − I) ⊆ N(T + I). (5)

(1.2) Show that V = R(T + I) + R(T − I). (6)

For more information. Email:

, (1.3) Show that V = R(T + I) ⊕ R(T − I). (5)
[16]



QUESTION 2

Consider the vector space V = C2 with scalar multiplication over the real numbers R, and let T : V → V be the linear
operator defined by
T (z1,z2) = (z1 − iz2,z2 − z2).
Use the Diagonalisability Test to explain whether or not T is diagonalisable. (Note that V is a vector space of
dimension 4 over R.)
[15]



QUESTION 3

Let fa,fb,fc ∈ P2(R) denote the Lagrange polynomials associated with the distinct real numbers a,b,c respectively. Let T :
P2(R) → P2(R) denote the projection on V = span{fa + fb,fb + fc} along W = span{fa + fc}.

(3.1) Find the matrix representation of T with respect to β = {fa,fb,fc}. (16)

(3.2) Find a formula for T(g) expressed as a linear combination of β where g ∈ P2(R). (8)
[24]



QUESTION 4

Let V be an inner product space over R with orthonormal basis β = {v1,v2,v3}, and let W = span{v1+v2,v2+v3}.

(4.1) Show that (5)

ha1v1 + a2v2 + a3v3,b1v1 + b2v2 + b3v3i = a1b1 + a2b2 + a3b3 for all a1,a2,a3,b1,b2,b3 ∈ R.


(4.2) Find a basis for W⊥ expressed in terms of β. (7)

(4.3) Find the vector in W closest to 3v2. (7)
[19]




[TURN OVER]



3 MAT3701
January/February 2021


QUESTION 5

Consider the inner product space P2 (R) over R with h·,·i defined by
For more information. Email: