UNIVERSITY EXAMINATIONS UNIVERSITEITSEKSAMENS
MAT3701 May/June 2020
LINEAR ALGEBRA III
Duration : 3 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.
Answer all the questions and show all calculations. Please read the questions carefully.
Since this is an open book examination, if the solution to a similar question is given, no marks will be awarded. The originality
of submissions will be verified by the software Turnitin.
TO UPLOAD YOUR ANSWER FILE
1. Access the myExams site at https://myexams.unisa.ac.za/portal and login using your student number and myUnisa password.
2. Click on the site MAT3701 Exam May 2020 (in the horizontal navigation bar at the top or click on the Sites button at the
top right-hand corner and select the site from the drop-down menu).
3. Once the site has loaded, click on eAssessment in the left navigation menu.
4. A page with the heading “Assignment List” will appear. Click on MAT3701 MayJune Exam 2020 under “Assignment Title”.
5. A new page will open. Under Attachments, click Choose file next to Select a file from computer. Now browse your
device for your answer file and select it for uploading. Remember, the file should be in pdf format and the name of the file
should be in the format Studentnumber MAT3701.
7. Once you have attached your answer file, the name of the file as well as the file size and upload time stamp will be displayed
under Attachments.
6. Tick the “Honour Pledge” button if you agree.
7. Click the “Submit” button.
2 MAT3701
May/June 2020
QUESTION 1
Let and consider the following subspaces of M2×2 (C) defined by
W1 = {X ∈ M2×2 (C) : AX = XA} and W2 = {X ∈ M2×2 (C) : AX = X}.
(1.1) Find a basis for W1. (8)
(1.2) Find a basis for W1 ∩ W2. (8)
(1.3) Explain whether M2×2 (C) = W1 ⊕ W2. (1)
[17]
For more information. Email:
, QUESTION 2
Let T : C3 → C3 be a linear operator such that T2 = T and dim(N(T)) = 2. Show there exists a basis β for C3
such that [ where b,c ∈ C.
[15]
QUESTION 3
Let
1
andw = 00 .
0
Let T : R4 → R4 be the linear operator defined by T(x) = Ax and let W be the T–cyclic subspace of R4 generated by w.
(3.1) Find the T–cyclic basis for W generated by w. (8)
(3.2) Find the characteristic polynomial of TW. (2)
(3.3) For each eigenvalue of TW, find a corresponding eigenvector expressed as a linear combination of the (8) T-
cyclic basis for W.
[18]
QUESTION 4
Consider the inner product space P2 (R) over R with h·,·i defined by
hg,hi = g (a)h(a) + g(b)h(b) + g (c)h(c)
where a, b and c are distinct real numbers. Let β = {fa, fb, fc} be the set of Lagrange polynomials associated with a, b
and c respectively, and let P : P2 (R) → P2 (R) be the orthogonal projection on W = span .
(4.1) Show that ha1fa + b1fb + c1fc,gi = a1g(a) + b1g(b) + c1g(c) for all a1,b1,c1 ∈ R and g ∈ P2(R). (6)
(4.2) Show that is orthonormal. (7)
(4.3) Find a formula for P(g) expressed as a linear combination of β. (7)
[20]
[TURN OVER]
3 MAT3701
May/June 2020
QUESTION 5
Let T : V → V be a linear operator on a finite-dimensional inner product space V over C.
(5.1) Define what is meant by the adjoint operator T∗ of T. (2)
(5.2) Define what is meant by a normal operator T. (2)
For more information. Email:
, (5.3) If T is normal, show that kT(v)k = kT∗(v)k for all v ∈ V . (5)
(5.4) Suppose V = C3 and T : C3 → C3 is defined by T (z1,z2,z3) = (z1 + iz2 − iz3,−iz1, iz1 + iz3). Find (9) a formula for T∗
(z1,z2,z3).
[18]
QUESTION 6
It is given that A ∈ M3×3 (C) is a normal matrix with eigenvalues 1 and −1 and corresponding eigenspaces
E1 = span
and
E−1 = span .
(6.1) Find the spectral decomposition of A. (11)
(6.2) Find A. (1)
[12]
TOTAL MARKS: [100]
c
UNISA 2020
For more information. Email:
MAT3701 May/June 2020
LINEAR ALGEBRA III
Duration : 3 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.
Answer all the questions and show all calculations. Please read the questions carefully.
Since this is an open book examination, if the solution to a similar question is given, no marks will be awarded. The originality
of submissions will be verified by the software Turnitin.
TO UPLOAD YOUR ANSWER FILE
1. Access the myExams site at https://myexams.unisa.ac.za/portal and login using your student number and myUnisa password.
2. Click on the site MAT3701 Exam May 2020 (in the horizontal navigation bar at the top or click on the Sites button at the
top right-hand corner and select the site from the drop-down menu).
3. Once the site has loaded, click on eAssessment in the left navigation menu.
4. A page with the heading “Assignment List” will appear. Click on MAT3701 MayJune Exam 2020 under “Assignment Title”.
5. A new page will open. Under Attachments, click Choose file next to Select a file from computer. Now browse your
device for your answer file and select it for uploading. Remember, the file should be in pdf format and the name of the file
should be in the format Studentnumber MAT3701.
7. Once you have attached your answer file, the name of the file as well as the file size and upload time stamp will be displayed
under Attachments.
6. Tick the “Honour Pledge” button if you agree.
7. Click the “Submit” button.
2 MAT3701
May/June 2020
QUESTION 1
Let and consider the following subspaces of M2×2 (C) defined by
W1 = {X ∈ M2×2 (C) : AX = XA} and W2 = {X ∈ M2×2 (C) : AX = X}.
(1.1) Find a basis for W1. (8)
(1.2) Find a basis for W1 ∩ W2. (8)
(1.3) Explain whether M2×2 (C) = W1 ⊕ W2. (1)
[17]
For more information. Email:
, QUESTION 2
Let T : C3 → C3 be a linear operator such that T2 = T and dim(N(T)) = 2. Show there exists a basis β for C3
such that [ where b,c ∈ C.
[15]
QUESTION 3
Let
1
andw = 00 .
0
Let T : R4 → R4 be the linear operator defined by T(x) = Ax and let W be the T–cyclic subspace of R4 generated by w.
(3.1) Find the T–cyclic basis for W generated by w. (8)
(3.2) Find the characteristic polynomial of TW. (2)
(3.3) For each eigenvalue of TW, find a corresponding eigenvector expressed as a linear combination of the (8) T-
cyclic basis for W.
[18]
QUESTION 4
Consider the inner product space P2 (R) over R with h·,·i defined by
hg,hi = g (a)h(a) + g(b)h(b) + g (c)h(c)
where a, b and c are distinct real numbers. Let β = {fa, fb, fc} be the set of Lagrange polynomials associated with a, b
and c respectively, and let P : P2 (R) → P2 (R) be the orthogonal projection on W = span .
(4.1) Show that ha1fa + b1fb + c1fc,gi = a1g(a) + b1g(b) + c1g(c) for all a1,b1,c1 ∈ R and g ∈ P2(R). (6)
(4.2) Show that is orthonormal. (7)
(4.3) Find a formula for P(g) expressed as a linear combination of β. (7)
[20]
[TURN OVER]
3 MAT3701
May/June 2020
QUESTION 5
Let T : V → V be a linear operator on a finite-dimensional inner product space V over C.
(5.1) Define what is meant by the adjoint operator T∗ of T. (2)
(5.2) Define what is meant by a normal operator T. (2)
For more information. Email:
, (5.3) If T is normal, show that kT(v)k = kT∗(v)k for all v ∈ V . (5)
(5.4) Suppose V = C3 and T : C3 → C3 is defined by T (z1,z2,z3) = (z1 + iz2 − iz3,−iz1, iz1 + iz3). Find (9) a formula for T∗
(z1,z2,z3).
[18]
QUESTION 6
It is given that A ∈ M3×3 (C) is a normal matrix with eigenvalues 1 and −1 and corresponding eigenspaces
E1 = span
and
E−1 = span .
(6.1) Find the spectral decomposition of A. (11)
(6.2) Find A. (1)
[12]
TOTAL MARKS: [100]
c
UNISA 2020
For more information. Email:
