# Linear algebra

## Literature

• [Geil] Olav Geil, "Elementary Linear Algebra". Pearson, 2015. ISBN: 978-1-78448-372-2.

## 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

## 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 $2×2$ 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.
• 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 $v$ 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 linear combination 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.
• $T:X\to Y$ 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 $T:{ℝ}^{n}\to {ℝ}^{m}$. 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?
• $f:ℝ\to ℝ$, $f\left(x\right)={x}^{2}+1$
• $g:ℝ\to ℝ$, $g\left(x\right)={x}^{3}+1$
• $h\left(x\right)$ is the CPR-number for $x$.
• 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 $T$ er linear and $T\left(v\right)$ is known, what is $T\left(cv\right)$. 57
• Determine, if $T:{ℝ}^{n}\to {ℝ}^{m}$ 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 $U\circ T$ from rules for $U$ and $T$. 69. Find standard matrices for $T$, $U$ and $U\circ T$. 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 $B={A}^{-1}$. 1,3
• Given ${A}^{-1}$ and ${B}^{-1}$. Find the inverse of combinations of $A$ and $B$. 9, 11.
• Elementary matrices. Find inverses. 17, 19. Given $A$, $B$, 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.
• Row reduction to calculate ${A}^{-1}B$. 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 $R$ of $A$ and a $P$ such that $PR=A$. 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 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 $2×2$ 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 $det\left(AB\right)=det\left(A\right)det\left(B\right)$ for $2×2$ matrices. 71
• Section 3.2 Prove that $det\left({B}^{-1}AB\right)=det\left(A\right)$ 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, 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 $A$ and the null space of ${A}^{T}$
• When $A$ 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 $v$ given ${\left[v\right]}_{\mathsc{ℬ}}$ and $\mathsc{ℬ}$. 1, 7
• Given $v$ as a linear combination of $\mathsc{ℬ}$, what is ${\left[v\right]}_{\mathsc{ℬ}}$? 13
• Find ${\left[v\right]}_{\mathsc{ℬ}}$ given $\mathsc{ℬ}$ 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 $\left[{\left[{e}_{1}\right]}_{\mathsc{ℬ}}{\left[{e}_{2}\right]}_{\mathsc{ℬ}}\right]$ and the matrix whose columns are the vectors in $\mathsc{ℬ}$. 51, 53
• A basis $\mathsc{ℬ}$ for the plane is constructed by rotating the standard basis. What is the connection between $v$ and ${\left[v\right]}_{\mathsc{ℬ}}$. 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. ${\left[v\right]}_{A}={\left[v\right]}_{B}$? 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. $\mathsc{ℬ}$. 1,3,7
• Find the standard matrix for $T$ given ${\left[T\right]}_{\mathsc{ℬ}}$ and $\mathsc{ℬ}$. 11, 15
• Test your understanding of matrix representations of linear transformations 20-23, 25-38
• Find ${\left[T\right]}_{\mathsc{ℬ}}$, the standard matrix for $T$ and a rule for $T$ given $T\left({b}_{i}\right)$ for all $b\in \mathsc{ℬ}$. 47, 49, 51
• Find ${\left[T\right]}_{\mathsc{ℬ}}$ from $T\left({b}_{i}\right)$ as a linear combination of $\mathsc{ℬ}$. Then ﬁnd $T\left(w\right)$, where $w$ is a linear combination of $\mathsc{ℬ}$. 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 $2×2$ 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 for $B$ and $cB$ 81.
• Connection between eigenvalues (and eigenvectors?) for $B$ and ${B}^{T}$ 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 diagonal matrix $D$, s.t. $A=PD{P}^{-1}$ 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 ${A}^{k}$. 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 ${y}_{1}\left(0\right)=1$ og ${y}_{2}\left(0\right)=4$.(Solution: ${y}_{1}\left(t\right)=-{e}^{-3t}+2{e}^{4t}$. ${y}_{2}\left(t\right)=3{e}^{-3t}+{e}^{4t}$)
• 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
• $QR$-factorization. 25,27,29, 31
• Solve systems of equations using $QR$-factorization. 33, 35, 37,39 OBS: Show that the solutions you found to $Rx={Q}^{T}b$ are solutions to $Ax=b$. (An extra challenge: Why is this necessary.)
• Test your understanding of Gram-Schmidt and $QR$-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 $u$ as a sum $u=w+z$, where $w\in W$ and $z\in {W}^{\perp }$. 9,11
• As above. Moreover, ﬁnd the matrix ${P}_{W}$ 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 and orthogonal complement. 33-56.
• What is the orthogonal complement to the orthogonal complement? 63
• What is ${\left({P}_{W}\right)}^{2}$ and ${\left({P}_{W}\right)}^{T}$. 67
• Find ${P}_{W}$ 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
• Orthogonal matrices and eigenvalues. 49
• Let ${Q}_{x}$ and ${Q}_{z}$ be the matrices for a $9{0}^{\circ }$ rotation around the $x$-axis and the $z$-axis respectively. ${Q}_{x}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{3.26288pt}{0ex}}\phantom{\rule{3.26288pt}{0ex}}{Q}_{z}=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

Let $Q={Q}_{x}{Q}_{z}$ 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(${\left[1\phantom{\rule{3.26288pt}{0ex}}-1\phantom{\rule{3.26288pt}{0ex}}\phantom{\rule{3.26288pt}{0ex}}1\right]}^{T}$))

• 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 can 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

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

### 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.

## Old exams

Note: new structure in the organisation of the exam. Relevant from spring 2016 and onwards.
• 2019 autumn
• 2019 spring
• 2018 autumn
• 2018 spring
• 2017 autumn
• 2017 spring
• 2016 autumn
• 2016 spring
• Test set

## 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.

### Aalborg (email: dhaug16@student.aau.dk)

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

• Tuesday 22/10-19 16:15-17:45 in Auditorium 1.
• Tuesday 29/10-19 16:15-17:45 in Auditorium 1.
• Wednesday 6/11-19 16:15-17:45 in Auditorium 1.
• Tuesday 12/11-19 16:15-17:45 in Auditorium 1.
• Tuesday 19/11-19 16:15-17:45 in A309, Strandvejen 12–14.
• Tuesday 26/11-19 16:15-17:45 in A309, Strandvejen 12–14.
• Tuesday 3/12-19 16:15-17:45 in A309, Strandvejen 12–14.
• Tuesday 10/12-19 16:15-17:45 in A309, Strandvejen 12–14.
• Tuesday 17/12-19 16:15-17:45 in A309, Strandvejen 12–14.

## 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.

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 7 on Friday the 10th of January and Monday the 13th of January, both days at 9:00–12:00 and 12:30–15:30.

Before the re-exam there will be a Q&A-session on Monday the 24th of February at 8:00–12:00. This takes place in Auditorium 3.