Linear algebra

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

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.

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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 aquire 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 aquiring 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 og 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. Scalarproduct 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 2 × 2 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.
    • Rowoperations: 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 genral 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 consisten. 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 v in Span( S)?. 1,3,7
    • Is v in Span(S)? A coordinate in v is unknown. 17, 19
    • Is Ax = b consistent for all b? 31,33.
    • Test your understanding of span. 45-64.
    • About the connection between Span(S) and the span of a linearcombination of S. 71, 72. Consequences for row-operations: 77, 78.
  • Section 1.4:
    • Systems of equations where a coefficient r is unknown. For which values of r 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 S, with the same span as S.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 r. For which values of r, 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.
    • T : X Y is induced by a matrix. Find X and Y . 1, 3
    • Find the image of a vector under a linear transformation induced by a matrix. 7, 11
    • From the rule for T, find n and m, such that T : n m. 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?
      • f : , f(x) = x2 + 1
      • g : , g(x) = x3 + 1
      • h : The set of Danish citizens h(x) is the CPR-number for x.
      • 61, 65.
    • Determine by finding a spanning set of the null space, wheter 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 T er linear and T(v) is known, what is T(cv). 57
    • Determine, if T : n m 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, m × n, 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 U T from rules for U og T. 69. Find standard matrices for T, U og U T. 70, 71,72.
    • Test your understanding of section 2.8 - composition of linear transformations and their matrices. 56-58.
  • MatLab: Section 2.1 opg. 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 B = A1. 1,3
    • Given A1 and B1. Find the inverse of combinations of A and B. 9, 11.
    • Elementary matrices. Find inverses. 17, 19. Givet A, B, find elementary matrices E, such that EA = B. 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.
    • Rowreduction to calculate A1B. 19
    • Test your understanding of Section 2.4. 35-54.
    • Solve a system of linear equations by inverting the coefficient matrix. 57.
    • Rowreduction to determine reduced row echelon form R of A and a P s.t. PR = A. 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 invers (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 2 × 2 matrix. 1, 3, 7. Do the calculation using the formula on p. 200.
    • Determinant of a 3 × 3 matrix using cofactors. 13, 15
    • Calculate determinants - choose your preferred method. 21, 23.
    • Determinant of 2 × 2 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 det(AB) = det(A)det(B) for 2 × 2 matrices. 71
  • Section 3.2 Prove that det(B1AB) = det(A) for n × n matrices A and B, where B 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, nullspace, 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 A and the null space of AT
      • When A is on reduced echelon form. 1, 3.
      • In general. 7.
    • Find the dimension of a subspace. 15
    • Find en basis for rækkerum. 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 generating. 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: Coordinatesystems. Section 4.4.

Exercises:

  • Section 4.4.
    • Find v given [v] and . 1, 7
    • Given v as a linear combination of , what is [v]? 13
    • Find [v] given and v. 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 [[e1][e2]] 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 v and [v]. 55, 67, 75
    • Equations for cone sections before and after change of basis. 79
    • What does it imply, that there is a vector v, s.t. [v]A = [v]B? 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 T wrt. . 1,3,7
    • Find the standard matrix for T given [T] and . 11, 15
    • Test your understanding of matrixrepresentations of linear transformations 20-23, 25-38
    • Find [T], the standardmatrix for T and a rule for T given T(bi) for all b . 47, 49, 51
    • Find [T] udfra T(bi) as a linearcombination of . Then find T(w), where w is a linearcombination of . 39, 55 43,59

Hand-in exercises: 7, 15, 39, 47 from Chapter 4.5.

13. session:

Topic: Eigenvectors og og eigenvalues. Sections 5.1 and 5.2 util 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 2 × 2 matrix have any (reat) eigenvalues? 41
    • Test your understanding of characteristic polynomial, multiplicity of eigenvalues. 53-59, 61,63-65, 69-72.
    • Connection between eigenspaces for B og cB 81.
    • Connection between eigenvalues (and egenvectors?) for B og BT 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 A and the characteristic polynomial. Find P and a diagonalmatrix D, s.t. A = PDP1 or explain why A 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 A is diagonalizable. 49, 51
    • Given eigenvalues and a basis for the eigenspaces, find Ak. 57, 59
    • Given a matrix and the characteristic polynomial. One entrance is an unkonown. 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: Ortogonality, Gram Schmidt, QR-faktorization. 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 y1(0) = 1 og y2(0) = 4.(Solution: y1(t) = e3t + 2e4t. y2(t) = 3e3t + e4t)
  • 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
    • QR-faktorization. 25,27,29, 31
    • Solve systems of equations using QR-faktorization. 33, 35, 37,39 OBS: Show that the solutions you found to Rx = QT b are solutions to Ax = b. (An extra challenge: Why is this necessary.)
    • Test your understanding of Gram-Schmidt and QR-faktorization. 41-52

Hand-in exercises: 9, 25, 33 from Chapter 6.2.

16. session:

Topic: Ortogonale projektioner. 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 u as a sum u = w + z, where w W and z W. 9,11
    • As above. Moreover, find the matrix PW for orthogonal projection on W, find the distance to W. 17,19,21 Hint to 21: Warning - the columns of A are not linearly independent.
    • Test your understanding of orthogonal projection og orthogonal complement. 33-56.
    • What is the orthogonal complement to the orthogonal complement? 63
    • What is (PW )2 and (PW )T . 67
    • Find PW given an orthonormal basis for W. 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:

  • Section 6.5
    • Recognize an orthogonal matrix. 1,4,5,3
    • Decide, if an orthogonal 2 × 2 matrix is a reflection or a rotation and determine the line of reflection or the angle of rotation. 9, 11
    • Orthogonale matrices and eigenvalues. 49
    • Let Qx and Qz be the matrices for a 90 rotation around the x-axis and the z-axis respectively. Qx = 10 0 0 0 1 0 1 0 Qz = 0 10 1 0 0 0 0 1

      Let Q = QxQz be the matrix for the combined transformation. This is a rotation too. Find the eigenspace associated to the eigenvalue 1 and hence the axis of rotation. (Answer: Span([1 11]T ))

  • Notice the different ways of posing multiple choice problems.

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

Self-study session 1

Self-study session 1 is supported by screencast 2 and 3 that are available in the MATLAB center.

Exercise 3 at page 90 mentions the function rotdeg. This function kan be downloaded here.

Literature: Appendix D

Self-study session 2

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

Click here to download the Matlab code mentioned in the pdf.. Note, that it is 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

Self-study session 4

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.

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 24 hours 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 recieved a response!. Please indicate in the email what you need help with (typically jst 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.

Aalborg (email: tmort15@student.aau.dk)

Currently the allocated dates if you have signed up by email are (will be updated throughout the semester):

  • Monday 8/10-18 16:15-17:45 in Auditorium 1.
  • Wednesday 10/10-18 16:15-17:45 in Auditorium 1.
  • Wednesday 17/10-18 16:15-17:45 in Auditorium 1.
  • Thursday 18/10-18 16:15-17:45 in Auditorium 1.
  • Monday 22/10-18 16:15-17:45 in Auditorium 1.
  • Friday 26/10-18 16:15-17:45 in Auditorium 1.
  • Monday 29/10-18 16:15-17:45 in Auditorium 1.
  • Thursday 1/11-18 16:15-17:45 in Auditorium 1.
  • Monday 5/11-18 16:15-17:45 in Auditorium 1.
  • Wednesday 7/11-18 16:15-17:45 in Auditorium 1.
  • Monday 12/11-18 16:15-17:45 in Auditorium 1.
  • Wednesday 14/11-18 16:15-17:45 in Auditorium 1.
  • Tuesday 20/11-18 16:15-17:45 in Auditorium 1.
  • Thursday 22/11-18 16:15-17:45 in Auditorium 1.
  • Monday 26/11-18 16:15-17:45 in Auditorium 1.
  • Thursday 29/11-18 16:15-17:45 in Auditorium 1.
  • Monday 3/12-18 16:15-17:45 in Auditorium 1.
  • Wednesday 5/12-18 16:15-17:45 in Auditorium 1.
  • Tuesday 11/12-18 16:15-17:45 in Auditorium 1.
  • Thursday 13/12-18 16:15-17:45 in Auditorium 1.

Esbjerg (email: baq@civil.aau.dk)

Here, the math cafe generally runs Wednesday afternoon.
Currently the allocated dates if you have signed up by email are (will be updated throughout the semester):

  • Wednesday 17/10-18 16:15-17:45 in room B206.
  • Wednesday 24/10-18 16:15-17:45 in room B206.
  • Wednesday 31/10-18 16:15-17:45 in room B206.
  • Wednesday 7/11-18 16:15-17:45 in room B206.
  • Wednesday 14/11-18 16:15-17:45 in room B206.
  • Wednesday 21/11-18 16:15-17:45 in room B206.
  • Wednesday 28/11-18 16:15-17:45 in room B206.
  • Wednesday 5/12-18 16:15-17:45 in room B206.
  • Wednesday 12/12-18 16:15-17:45 in room B206.
  • Wednesday 19/12-18 16:15-17:45 in room B206.

Copenhagen (email: roenby@math.aau.dk)

Here, the math cafe generally runs Monday afternoon
Currently the allocated dates if you have signed up by email are (will be updated throughout the semester):

  • Monday 17/9-18 16:15-17:45 in room 0.108, FKJ10A.
  • Monday 24/9-18 16:15-17:45 in room 0.108, FKJ10A.
  • Monday 1/10-18 16:15-17:45 in room 0.108, FKJ10A.
  • Monday 29/10-18 16:15-17:45 in room 0.108, FKJ10A.
  • Monday 5/11-18 16:15-17:45 in room 0.108, FKJ10A.
  • Monday 12/11-18 16:15-17:45 in room 0.108, FKJ10A.
  • Monday 19/11-18 16:15-17:45 in room 0.108, FKJ10A.
  • Monday 26/11-18 16:15-17:45 in room 0.108, FKJ10A.
  • Monday 3/12-18 16:15-17:45 in room 0.108, FKJ10A.
  • Wednesday 12/12-18 16:15-17:45 in room 0.108, FKJ10A.

Maths-event

Do you both want to improve your maths skills before the exam and also see how the maths at the first study year can be applied?

Then Maths Saturday is just what you need. The event is held as a workshop in both Aalborg and Copenhagen on the following dates:

  • Aalborg: 10th of November 2018 at 9:30-15:00. Auditorium 1, Badehusvej
  • Copenhagen: 17th of November 2018 at 9:30-15:00. Room 0.108 in building D, FKJ10A

The day will consist of two mini-projects where the teacher will give a short presentation of each subject (one before lunch and one after), and afterwards the teacher will assist you as needed during the project. Through both "pen and paper" exercises and MATLAB exercises the projects will strengthen your maths skills. Hence, this is a great occasion to practice linear algebra and prepare for the exam.

It is possible to participate as non-Danish speaker since the course material and exercises will be in English, but the short intro by the teacher will be held in Danish.

A free sandwich is served for lunch and therefore you need to sign up by filling out the form below no later than Tuesday the 6th of November 2018.

The sign-up for the event has been closed.

Mini-project 1: Image representation

In this mini-project, we will see how images can be represented using matrices, and how linear algebra can be used to transform images in various ways. For instance, it can be used to mirror an image along a line and to perform rotations and translations.

We will examine different transformations of images, and during the exercises MATLAB will naturally present itself as a tool to try out such transformations in practice. Additionally, each exercise contains a theoretical aspect through the explanation of the specific type of mathematical transformation needed. Here, we will need pen and paper, as well as sound mathematical arguments.

For this miniproject, the following material is used.

Mini-project 2: Computer graphics and planetary orbits

The topic of this mini-project is the application of linear algebra to computer graphics and the orbits of planets.

Assuming that planetary orbits are circular, they can be described by representing the position of the planetby a vector, and then repeatedly applying a rotation matrix to obtain the new positions on the orbit. We will create a graphical simulation of the planets orbit around the sun by showing several frames per second, each frame containing a plot of the position vector multiplied by the rotation matrix. The next step is to add a satellite orbiting the planet by using similar techniques. In this case, affine rotation matrices are used, meaning that the centre of rotation changes over time.

In the mini-project you will get a chance to work with affine rotation matrices, their geometric interpretation, and see how to use MATLAB to simulate the movement of an object.

For this miniproject, the following material is used.

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.

Dates for Q&A-sessions

We offer assistance with the exam preparation in both calculus and linear algebra at all three campi. This consists of a Q&A-session, where it is possible to ask questions within the syllabus and receive help in solving concrete exercises. During this session, it is also possible solve exercises on your own, and then ask for hints if you get stuck. The session takes as its starting point the old exam questions, which may be found here at first.math.aau.dk. We recommend that you solve as many as you can beforehand, such that you know where you come short. Note that the teaching assistants will not visit you in your group rooms. Instead, everyone will be solving exercises individually or in small groups in the rooms specified below.

Aalborg

Teaching assistants will be available to help you while you prepare for the exam. They are present in AUD 6 and 7 on Friday the 11th of January at 12:00-15:00 and Saturday the 12th of January at 8:00-11:00.

Before the re-exam there will be a Q&A-session on Monday the 18th of February. This takes place in AUD 1 and 2 at 16:00–19:00.

Copenhagen

There will be a Q&A-session Thursday the 10th of January at 8:15-10:00. This takes place in 0.108, FKJ10A in building D.. The following day, Friday the 11th of January at 8:15-12:00, there will be a repetition lecture in 0.06, FKJ12 in building B.

Before the re-exam there will be a Q&A-session on Friday the 15th of February at 13:00–15:00 (Note the change of schedule). This takes place in 3.152 at FKJ10A, building D.