Math cafe
Please remember that we have a math-cafe where you can get help with your unsolved exercises (provided you have signed up as described here). Next session is:
- Aalborg: Not scheduled yet.
- Esbjerg: Not scheduled yet.
- Copenhagen: Not scheduled yet.
Literature
- [Geil] Olav Geil, "Elementary Linear Algebra". Pearson, 2015. ISBN: 978-1-78448-372-2.
Supplementary literature
MATLAB
The use of Matlab is an integral part of the four sessions without lectures (mini-projects) and, up to some extent, in other sessions as well. Students can freely download Matlab via the ICT link at http://www.tnb.aau.dk/. One can find more information in the MATLAB center (including a video showing how to install it).
Exam
The course is evaluated through a four hour written exam without the use of any electronic device. One may bring any kind of notes and books. For further information, see the tab Exam information
Bemærk
The information on this page does not apply to students from the programmes Byggeri & Anlæg in Aalborg and Maskin & Produktion in Aalborg. They can find course materials and exam information on Moodle instead.
Hand-ins
During the course, written exercises will be given. For the degree programs listed below, the enrolled students can only attend the exam if at least 10 out of 18 of these hand-ins are approved. The extent of each exercise is expected to be around one handwritten sheet of A4-paper.
If the degree programme is not listed, it is still possible to hand-in exercises and receive feedback.
The hand-in exercises for each lecture will be listed at the
Course Plan page.
- Applied Idustrial Electronics (Esbjerg)
- Biologi (Aalborg)
- Bioteknologi (Aalborg)
- Bygge- og anlægskonstruktion (Esbjerg)
- Byggeri og anlæg (Esbjerg)
- Bæredygtig energiteknik (Aalborg)
- Chemical Engineering and Biotechnology (Esbjerg)
- Eksportteknologi (Aalborg)
- Energi (Aalborg)
- Energi (Esbjerg)
- Energy engineering (Aalborg)
- Fysik (Aalborg)
- Globale forretningssystemer (Aalborg)
- Kemi (Aalborg)
- Kemi og bioteknologi (Aalborg)
- Kemi og bioteknologi (Esbjerg)
- Kemiteknologi (Aalborg)
- Manufacturing and Operations Engineering (København)
- Maskinkonstruktion (Esbjerg)
- Maskinteknik (Aalborg)
- Maskinteknik (Esbjerg)
- Matematik (Aalborg)
- Matematik-teknologi (Aalborg)
- Matematik-økonomi (Aalborg)
- Miljøvidenskab (Aalborg)
- Nanoteknologi (Aalborg)
- Sustainable Biotechnology (København)
Plan
Manual for the exercises:
- Exercises are structured according to content.
- First, do the exercises that are bold. Then do the rest.
- In general, each student is responsible for doing enough
exercises to acquire basic skills and routine. Some students need
many exercises to get this, others fewer.
- Skills from one session will often be a prerequisite for the next
sessions. Hence, it is very important to keep up and master the
skills. Otherwise, one may have to spend a lot of time during
a later session practising skills which should have been routine
by then.
- Not only inquiring basic skills, but also understanding the text is
important. Hence, the exercises testing understanding should be
taken seriously. When using mathematical techniques, it is of fundamental
importance to know why and when a given method can be applied.
1. session:
Topic: Introduction to vectors and matrices. Sections 1.1, 6.1 pp. 361-366. However, on pp. 364-365 read the theorems only. Section 1.2 until the bottom of p. 19.
Exercises:
- Section 1.1 Matrices and vectors
- Addition and multiplication by a scalar. 1,3,7.
- Transposition. 5,11,9.
- Is it possible to add two matrices: 19, 21,
- Test your understanding of matrices and vectors:
37-39, 41,42, 44-56.
- Section 6.1. Scalar-product and Orthogonality.
- Calculate norm of and distance between vectors 1, 7.
- Are two vectors orthogonal: 9, 15
- Section 1.2
- Matrix-vector product: 1,3,5,7 9,11,15. Hint:
Pencast.
- Express a vector as a linear combination of a set of
vectors.: 29, 33, 31, 35, 39
- Test your understanding of linear combinations.
45-51.
- Section 1.1
- Determine rows and columns in a matrix 29, 31
- Symmetric matrices 71, 72, 75.
- Skew matrices 79, 80, 81
Hand-in exercises: 7 from Chapter 1.2; 1, 9 from Chapter 6.1.
2. session:
Topic: Matrix-vector product and systems of linear equations. Sections 1.2 from p. 19, 1.3.
Exercises:
- Section 1.2.
- Write
rotation matrices. 17, 19
- Test your understanding of matrix-vector products.
51-64
- Section 1.3.
- Write the coefficient matrix and the augmented
matrix of a linear system: 1,3,5.
- Row operations: 7,9,11
- Decide if a vector is a solution to a system of linear
equations. 23, 25.
- Decide from the reduced echelon form, if a system of
linear equations is consistent. If so, find the general
solution. 39, 43, 41.
- As above, but furthermore write the general solution
in vector form. 47, 49.
- Test your understanding of systems of linear
equations and their matrices. 57-76
Hand-in exercises: 17 from Chapter 1.2; 23 from Chapter 1.3.
3. session:
Topic: Gauss-elimination. Span. Sections 1.4 and 1.6
Exercises:
- Section 1.4:
- Decide, if a linear system is consistent. If so, find the
general solution. 1,5,9,3,7,11
- Determine rank and nullity of a matrix. 37, 35.
- Test your understanding of Gauss-elimination: 53-72.
- Section 1.6.
- Is
in Span( )?.
1,3,7
- Is
in Span()?
A coordinate in
is unknown. 17, 19
- Is
consistent for all ?
31,33.
- Test your understanding of span. 45-64.
- About the connection between Span()
and the span of a linear combination of .
71, 72. Consequences for row-operations: 77, 78.
- Section 1.4:
- Systems of equations where a coefficient
is unknown. For which values of
is the system inconsistent. 17, 19,21
Hand-in exercises: 5, 37 from Chapter 1.4; 17 from Chapter 1.6.
4. session:
Topic: Linear independence. Section 1.7.
Exercises:
- Section 1.7.
- Determine, if a set of vectors is linearly dependent.
1,5,7,9,11
- Find a small subset of ,
with the same span as . 13,
15.
- Determine, if a set of vectors is linearly independent.
23,25,27
- Test your understanding of linear (in)dependence 1.7
63-82.
- Given a set of vectors, one of which has an unknown
coordinate .
For which values of ,
if any, is the set linearly dependent. 41.
Hand-in exercises: 23, 41 from Chapter 1.7.
5. session:
Topic: Linear transformations and matrices. Sections 2.7, 2.8 until the middle of p. 185. (For functions in general (injectivity, surjectivity, and bijectivity), see Appendix B)
Exercises:
- Section 2.7.
-
is induced by a matrix. Find
and .
1, 3
- Find the image of a vector under a linear
transformation induced by a matrix. 7, 11
- From the rule for ,
find
and ,
such that .
21 23
- Find the standard matrix of a linear transformation.
25, 27, 29,31, 33
- Test your understanding of linear transformations
and their matrix representations. 35-54.
- Section 2.8.
- Find a generating set for the range. 1,3
- Are the following maps surjective (onto), injective
(one-to-one), bijective?
- ,
- ,
-
is the CPR-number for .
- 61, 65.
- Determine by finding a spanning set of the null
space, whether a transformation is injective. 13,
15, 17
- Determine by finding the standard matrix, whether a
linear transformation is injective. 25, 29, surjective. 33,
35.
- Test your understanding of section 2.8 (till p. 185).
41-55.
- Section 2.7.
- If
er linear and
is known, what is .
57
- Determine, if
is linear. 77, 73, 79
Hand-in exercises: 3, 7, 79 from Chapter 2.7; 27 from Chapter 2.8.
6. session:
Topic: Matrix multiplication, composition of linear transformations. Sections 2.1 and 2.8. From the middle of p. 185 until p. 187.
Exercises:
- Section 2.1.
- If the product of two matrices is defined, find the size,
,
of the product. 1,3
- Calculate matrix products. 5,9,11,7. Calculate a
given entrance in a product matrix. 25
- Test your understanding of the matrix product.
33-50.
- Section 2.8.
- Find a rule for
from rules for
and .
69. Find standard matrices for ,
and .
70, 71,72.
- Test your understanding of section 2.8 - composition
of linear transformations and their matrices. 56-58.
- MatLab: Section 2.1 exercise 53
Hand-in exercises: 15 from Chapter 2.1; 69, 70 from Chapter 2.8.
7. session:
Topic: Invertible matrices and invertible linear transformations. Sections 2.3, 2.4, and 2.8, pp. 187-188.
Exercises:
- Section 2.3.
- determine whether .
1,3
- Given
and .
Find the inverse of combinations of
and .
9, 11.
- Elementary matrices. Find inverses. 17, 19. Given
,
,
find elementary matrices ,
such that .
25, 29.
- Section 2.4. Is a given matrix invertible? If so, find the inverse.
1, 3, 5, 9, 13
- Section 2.8 The connection between invertible matrices and
invertible linear transformations. 59,60.
- Section 2.4.
- Row reduction to calculate .
19
- Test your understanding of Section 2.4. 35-54.
- Solve a system of linear equations by inverting the
coefficient matrix. 57.
- Row reduction to determine the reduced row echelon form
of
and a
such that .
27
- Section 2.3
- The column correspondence property. 67.
- Write a column as a linear combination of the pivot
columns. 75.
- MatLab. Section 2.8. Find the standard matrix for a
linear transformations calculate the inverse (MatLab)
Use this to find a rule for the inverse transformation.
100
Hand-in exercises: 67 from Chapter 2.3; 19, 57 from Chapter 2.4.
8. session:
Topic: Determinants. Sections 3.1 and 3.2 until p. 217, l. 9.
Exercises:
- Section 3.1
- Determinant of a
matrix. 1, 3, 7. Do the calculation using the formula
on p. 200.
- Determinant of a
matrix using cofactors. 13, 15
- Calculate determinants - choose your preferred
method. 21, 23.
- Determinant of
matrices and area. 29
- Determinant and invertibility. 37.
- Test your understanding of determinants and
cofactors. 45-64
- Section 3.2
- Calculate determinants- develop after a given column
1, 5
- Calculate determinants using row-operations . 13, 15,
21, 23
- Test your understanding of the properties of
determinants. 39-58.
- Section 3.1 Prove that
for
matrices. 71
- Section 3.2 Prove that
for
matrices
and ,
where
is invertible. 71
Hand-in exercises: 23, 26, 38 from Chapter 3.1; 13 from Chapter 3.2.
9. session:
Topic: Subspaces, basis for subspaces. Sections 4.1 and 4.2 until the middle of p. 245.
Exercises:
- Section 4.1
- Find a generating set for a subspace. 1, 5, 9.
- Is a vector in the null space of a given matrix. 11, 15
- Is a vector in the column space of a given matrix.
19,21
- Find a generating set for the null space of a matrix.
27, 29
- Test your understanding of subspace, null-space,
column space. 43-62.
- Prove that a set is not a subspace. 81,
- Prove that a set is a subspace. 89
- The null space of a linear transformation is a
subspace. 96.
- Section 4.2.
- Find a basis for the null space and column space of
a matrix. 1, 3, 5.
- Find a basis for the null space and range of a linear
transformation. 9
- Section 4.1 Find a generating set for the column space of a
matrix. With a prescribed number of elements. 67,69.
Hand-in exercises: 11, 21, 81 from Chapter 4.1; 1 from Chapter 4.2.
10. session:
Topic: Dimension, Rank and nullity. The remaining parts of 4.2, 4.3.
Exercises:
- Section 4.2
- Find a basis for the range and null space of a linear
transformation. 9, 11, 13 15
- Find a basis for a subspace 17, 19, 23
- Test your understanding of basis and dimension.
33-52.
- Section 4.3.
- Find the dimension of the column space,
null space and row space of a matrix
and the null space of
- When
is on reduced echelon form. 1, 3.
- In general. 7.
- Find the dimension of a subspace. 15
- Find en basis for a row space. 17, 19.
- Test your understanding of dimension of subspaces
connected to matrices. 41-60.
- Prove that a given set is a basis for a given subspace. 61,
63.
- Section 4.2
- Explain why a set is not a generating set. 55
- Explain why a set is not linearly independent. 57.
Hand-in exercises: 9, 23 from Chapter 4.2; 1, 7 from Chapter 4.3.
11. session:
Topic: Coordinate systems. Section 4.4.
Exercises:
- Section 4.4.
- Find
given
and .
1, 7
- Given
as a linear combination of ,
what is ?
13
- Find
given
and .
15, 17, 19
- Write a vector as a linear combination of a set of
vectors. 25, 27
- Test your understanding of coordinate systems. 31-50
- What is the connection between the matrix
and the matrix whose columns are the vectors in
.
51, 53
- A basis
for the plane is constructed by rotating the standard
basis. What is the connection between
and .
55, 67, 75
- Equations for cone sections before and after change
of basis. 79
- What does it imply, that there is a vector ,
s.t. ?
99.
Hand-in exercises: 7, 23, 53 from Chapter 4.4.
12. session:
Topic: Linear transformations and coordinate systems. Section 4.5.
Exercises:
- Section 4.5
- Find the matrix for
wrt. .
1,3,7
- Find the standard matrix for
given
and .
11, 15
- Test your understanding of matrix representations of
linear transformations 20-23, 25-38
- Find ,
the standard matrix for
and a rule for
given
for all .
47, 49, 51
- Find
from
as a linear combination of .
Then find ,
where
is a linear combination of .
39, 55 43,59
Hand-in exercises: 7, 15, 39, 47 from Chapter 4.5.
13. session:
Topic: Eigenvectors and eigenvalues. Sections 5.1 and 5.2 until p.
307.
Exercises:
- Section 5.1
- Show that a vector is an eigenvector. 3, 7
- Show that a scalar is an eigenvalue. 13, 21
- Test your understanding of eigenvalues and
eigenvectors. 41-56, 57-60
- Section 5.2
- Find eigenvalues and a basis for the associated
eigenspaces
- For a matrix - given the characteristic
polynomial 1, 11
- For a matrix. 15, 19
- For a linear transformation - given the
characteristic polynomial. 31
- For a linear transformation. 37
- Does a
matrix have any (real) eigenvalues? 41
- Test your understanding of characteristic polynomial,
multiplicity of eigenvalues. 53-59, 61,63-65, 69-72.
- Connection between eigenspaces for
and
81.
- Connection between eigenvalues (and eigenvectors?) for
and
83.
Hand-in exercises: 3 from Chapter 5.1; 1, 15, 37 from Chapter 5.2.
14. session:
Topic: Diagonalization. Section 5.3
Exercises:
- Section 5.3
- Given a matrix
and the characteristic polynomial. Find
and a diagonal matrix ,
s.t.
or explain why
is not diagonalizable. 1, 3, 5,7,9
- As above, but the characteristic polynomial is not
given. 13, 15 17
- Test your understanding of diagonalization of
matrices. 29-37, 39-43, 45,46
- Determine from the eigenvalues and their multiplicity
whether
is diagonalizable. 49, 51
- Given eigenvalues and a basis for the eigenspaces,
find .
57, 59
- Given a matrix and the characteristic polynomial.
One entrance is an unknown. For which values is
the matrix not diagonalizable. 63
- Section 5.5. These exercises are connected to self-study session 3.
- Find the general solution to a system of differential
equations.. 45
Hand-in exercises: 7, 13, 17, 50 from Chapter 5.3.
15. session:
Topic: Orthogonality, Gram Schmidt, QR-factorization. Section 6.2.
Exercises:
- Section 5.5. These exercises are related to miniproject 3.
- Test your understanding of systems of linear
differential equations. 8-11
- In exercise 45, find the solution satisfying
og .(Solution:
.
)
- Section 6.1 (refresh your memory)
- Test your understanding of the inner product and
orthogonality. 61-70, 73-80
- Section 6.2
- Determine whether a set of vectors is orthogonal. 1,
3, 7
- Apply Gram-Schmidt. 9,11, 13,15
- -factorization.
25,27,29, 31
- Solve systems of equations using -factorization.
33, 35, 37,39 OBS: Show that the solutions you found
to
are solutions to .
(An extra challenge: Why is this necessary.)
- Test your understanding of Gram-Schmidt and -factorization.
41-52
Hand-in exercises: 9, 25, 33 from Chapter 6.2.
16. session:
Topic: Orthogonal projections. Section 6.3.
Exercises:
- Section 6.1 (refresh your memory) Projection on a line. 43,
45
- Section 6.3
- Find a basis for the orthogonal complement. 1, 3, 5
- write a vector
as a sum ,
where
and .
9,11
- As above. Moreover, find the matrix
for orthogonal projection on ,
find the distance to .
17,19,21 Hint to 21: Warning - the columns of
are not linearly independent.
- Test your understanding of orthogonal projection and
orthogonal complement. 33-56.
- What is the orthogonal complement to the
orthogonal complement? 63
- What is
and .
67
- Find
given an orthonormal basis for .
75
Hand-in exercises: 9, 17, 67 from Chapter 6.3.
17. session:
Topic: Orthogonal matrices. Orthogonal transformations in the plane. Section 6.5 until p. 419.
Exercises:
Hand-in exercises: 1, 5, 9, 11 from Chapter 6.5.
18. session:
Topic: Rigid motion. Section 6.5 pp. 419-421. Repetition – for instance by going through an old set of exam questions.
Overview of the course.
Suggestion: Use the problems from one of the exams as a point of
departure and explain in broad terms what to do in each of the
problems.
Exercises:
- Section 6.5
- Determine the matrix and vector of a rigid motion. 61, 62, 64
- Old exams.
Hand-in exercises: 61, 64 from Chapter 6.5.
Self-study sessions
As an aid for answering the various exercises and in order to document all your answers you can use the file Self-study-1.m. For elementary information concerning MATLAB you can read or skip through Appendix D in the book.
Self-study session 1 is supported by screencast 2 and 3 that are available in the MATLAB center.
Literature: Appendix D
Self-study session 2 is supported by screencast 4 that is only available in Danish (Danish MATLAB center) at the moment.
Literature: Appendix D
Self-study session 3
This self-study session has specialized exercises for certain study programmes. Click your programme in the list below to see the description of the self-study. If your programme does not appear in the list, you should solve the general exercises found under “None of the above”.
LAND
The self-study session (in Danish) considers the translation from ED50 to ETRS89.
ComTek, EIT, PDP, ROB og ST
The self-study session (in Danish) considers signal processing, and in the third exercise you will need the file speech.wav.
None of the above
Solve these exercises, where you will need the accompanying MATLAB code. Note, that the code is contained in a zip compressed archive consisting of 5 files.
Self-study session 3 is supported by screencast 6 that is available in the MATLAB center.
Literature: Appendix D
The following MATLAB files are used in the self-study session:
Self-study session 4 is supported by screencast 7 that is available in the MATLAB center.
Literature: Appendix D
Glossary
A glossary of linear algebra-terms used in English and Danish may be downloaded here.
Old exams
Note: new structure in the organisation of the exam. Relevant from
spring 2016 and onwards.
- 2020 spring
- 2019 autumn
- 2019 spring
- 2018 autumn
- 2018 spring
- 2017 autumn
- 2017 spring
- 2016 autumn
- 2016 spring
- Test set
Previous exams
- Test set (2015 autumn)
- Test sets
- 2010
- 2011
- 2012
- 2013
- 2014
- 2015 autumn
Curriculum
Literature:
- [Geil] Olav Geil, "Elementary Linear Algebra". Pearson, 2015. ISBN: 978-1-78448-372-2:
Curriculum ([Geil])::
- Section 1.1, 1.2, 1.3, 1.4, 1.6, 1.7
- Section 2.1, 2.3, 2.4, 2.7, 2.8
- Section 3.1, 3.2 to page 217 l.9
- Section 4.1, 4.2, 4.3, 4.4, 4.5
- Section 5.1, 5.2 to page 307 bottom, 5.3
- Orthogonality: Section 6.1 to page 366, 6.2, 6.3, 6.5.
- Appendix D
- Self-study sessions 1-4
Math cafe
Do you have a hard time understanding linear algebra and/or calculus at the first study year, and are you determined to do something about it?
Then Math cafe is just the right thing for you.
It is held throughout the semester at all three campuses (specific times and places are listed below).
It is an extra possibility for getting help with maths. A teaching assistant is available to help you with exercises from the last few lectures.
All you have to do is to sign up by sending an email to the assistant at least two days before the planned session. If the assistant hasn't received any email by that time Math Cafe is cancelled without further notice.
So you can only expect help if you have sent an email in due time and received a response!. Please indicate in the email what you need help with (typically just a specific exercise) without writing a long email about the details of you problem.
Note: This is an extra curricular activity, so it is NOT a valid excuse for not participating in other course activities or project work.
Information on when and where the math cafe will take place is coming soon.
Notice
Due to Covid-19, the math café is now digital. It is still necessary to contact dhaug16@student.aau.dk as described below. Then Danni can help on Thursdays 16:30–18:30, for instance on Discord.
When contacting Danni, it is important to be as specific as possible to allow him to prepare in advance for the café.
Currently the allocated dates if you have signed up by email are (will be updated throughout the semester):
- Thursday 13/2 16:30–18:30 in A309, Strandvejen 12–14.
- Thursday 20/2 16:30–18:30 in A309, Strandvejen 12–14.
- Thursday 27/2 16:30–18:30 in A315, Strandvejen 12–14.
- Thursday 5/3 16:30–18:30 in A309, Strandvejen 12–14.
- Thursday 12/3 16:30–18:30 in A309, Strandvejen 12–14.
- Thursday 19/3 16:30–18:30 in Auditorium 5, Badehusvej 13.
- Thursday 26/3 16:30–18:30 on Discord.
- Thursday 2/4 16:30–18:30 on Discord.
- Thursday 16/4 16:30–18:30 on Discord.
- Thursday 23/4 16:30–18:30 on Discord.
- Thursday 30/4 16:30–18:30 on Discord.
- Thursday 7/5 16:30–18:30 on Discord.
- Thursday 14/5 16:30–18:30 on Discord.
- Thursday 28/5 16:30–18:30 on Discord.
Currently the allocated dates if you have signed up by email are (will be updated throughout the semester):
- Thursday 26/3 16:30–18:30 on Discord.
- Thursday 2/4 16:30–18:30 on Discord.
- Thursday 16/4 16:30–18:30 on Discord.
- Thursday 23/4 16:30–18:30 on Discord.
- Thursday 30/4 16:30–18:30 on Discord.
- Thursday 7/5 16:30–18:30 on Discord.
- Thursday 14/5 16:30–18:30 on Discord.
- Thursday 28/5 16:30–18:30 on Discord.
Currently the allocated dates if you have signed up by email are (will be updated throughout the semester):
- Thursday 26/3 16:30–18:30 on Discord.
- Thursday 2/4 16:30–18:30 on Discord.
- Thursday 16/4 16:30–18:30 on Discord.
- Thursday 23/4 16:30–18:30 on Discord.
- Thursday 30/4 16:30–18:30 on Discord.
- Thursday 7/5 16:30–18:30 on Discord.
- Thursday 14/5 16:30–18:30 on Discord.
- Thursday 28/5 16:30–18:30 on Discord.
Notice
The information on this page does not apply to students from the programmes Byggeri & Anlæg in Aalborg and Maskin & Produktion in Aalborg.
Exam
The exam will be a digital exam with invigilation. That means that you have to show up like an ordinary written exam, but that the exam questions are answered online through Moodle.
All students must bring their own computer with internet access, but only the use of DigitalEksamen and Moodle is allowed – digital notes are not allowed. To prevent cheating the program ITX-Flex must be running during the exam. This must be installed in advance; how this is done can be found in the official guidelines.
We recommend using one of the following browsers to answer the questions in Moodle: Chrome, Firefox, Opera, or Safari. It is, in principle, possible to answer the questions using Internet Explorer or Edge, but the question layout may be inconvenient.
During the exam
At the start of the exam you are required to log in to both DigitalEksamen and ITX-Flex. Here, you will find a link to Moodle, where the exam questions themselves will be answered.
In Moodle, you are asked to choose between Danish and English exam questions – this can only be chosen once. It is a good idea to select a language before the exam in order to avoid delay on the exam day. Once the language has been selected, the corresponding exam questions will be unlocked at the start of the exam. Answer the questions like you would in any multiple-choice exam.
Submission
After having finished your attempt, you must first submit it in Moodle. Afterwards, you must download one of the forms found on the ‘Set of exam questions’ in DigitalEksamen/ITX-Flex. This is to be filled in with name and student-number and then uploaded and submitted in DigitalEksamen. This is important, as your hand-in cannot be graded otherwise. Once the submission in Moodle closes, you have an additional 10 minutes to finish your submission in DigitalEksamen.
What is allowed?
You are allowed to use handwritten, printed, and copied notes, as well as textbooks.
You are not allowed to use electronic devices, except for accessing DigitalEksamen and the exam page in Moodle. Visiting other webpages is not allowed either.
Additional information
For additional information about the exam and the current rules, we refer to the guidelines that may be found on the Moodle page for exams on the first year of study.
Preparation for the exam
The curriculum for the exam can be found under the tab "Curriculum", and the exercises at the exam will be within these topics. It is a good idea to cover the entire curriculum by using the overview of each lecture.
Example: The exercises about eigenvalues and eigenvectors are divided into:
- Section 5.1
- Show that a vector is an eigenvector. 3, 7
- Show that a scalar is an eigenvalue. 13, 21
- Test your understanding of eigenvalues and eigenvectors. 41-56, 57-60
- Section 5.2
- Find eigenvalues and a basis for the associated eigenspaces
- For a matrix - given the characteristic polynomial 1, 3,11
- For a matrix. 15, 19
- For linear transformation - given the characteristic polynomial. 31
- For en linear transformation. 37
- Does a $2 \times 2$ matrix have any (real) eigenvalues? 41
- Test your understanding of characteristic polynomial, multiplicity of eigenvalues. 53-59, 61,63-65, 69-72.
- Connection between eigenspaces of $B$ and $cB$ 81.
- Connection between eigenvalues (and eigenvectors?) of $B$ and $B^\top$ 83.
Reflect on the following general principles.
Which topics are connected/build upon others? Make an overview to yourself, and/or discuss it in your group.
Remember True/False.
Use these exercises to figure out the details of the curriculum.
Then solve previous exam questions - purpose: To see how the exercises are phrased. To practice the different types of multiple choice questions. Note that exam questions from previous exams which were not multiple choice can easily be relevant; the only difference is the way, the answer is given.
TA's during the exam preparation
We offer the help of TA's in your groups during the weeks before the exam. This is organized in MS Teams, where you sign up using the code found on the Moodle page for your course.
The help is available 9:00–16:30 on the five days before the exam including weekend. That is, starting Wednesday 10th June and ending Sunday 14th June.
This help consists of timeslots where you can ask questions within the topics of the course and receive help in solving concrete exercises. This will be based on the previous old exam sets, which may be found on these pages. We recommend to solve as many of these beforehand, such that you know where you come short.
Within the Team, you can create a new channel by clicking “More”/“Mere” (the three dots next to the Team name) and choosing “Add channel”/“Tilføj kanal”. Each group is asked to create one and only one channel named according to study programme and group number, e.g. “MATØK - B306”. When help is needed, go to the channel “General” and write “Hjælp @MATØK - B306” in the chat. By using “@” MS Teams creates a direct link to the group's channel which the TA can click when she or he is available.
During the period, the number of TA's will vary according to the expected need, so the waiting time may vary significantly. Please prepare well in advance to avoid finding yourselves at the end of a long queue on the final day before the exam.