# Linear algebra, 2016 autumn

## 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.

## 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. At the exam, there are multiple choice exercises along the lines of the True/False exercises in the textbook. These exercises count for 30$%$ of the points.
1. session:
Topic: Introduction to vectors and matrices.
• 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
2.session:
Topic: Matrix-vector product and systems of linear equations
• 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
3.session:
Topic: Gauss-elimination. Span.
• 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
4.session:
Topic: Linear independence.
• 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.
5.session:
Miniproject 1. Solve systems of linear equations using MatLab
6.session:
Topic: Linear transformations and matrices.
• Section 2.7.
• $T:X\to Y$ is induced by a matirx. 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, 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 $T\left(v\right)$ is known, what is $T\left(cv\right)$. 57
• Determine, if $T:{ℝ}^{n}\to {ℝ}^{m}$ is linear. 77, 73, 79

7.session:
Topic: Matrix multiplication, composition of linear transformations.
• 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 $U$ og $T$. 69. Find standard matrices for $T$, $U$ og $U\circ T$. 70, 71,72.
• Test your understanding of section 2.8 - composition of linear transformations and their matrices. 56-58.
• MatLab: Section 2.1 opg. 53
8.session:
Topic: Invertible matrices and invertible linear transformations.
• 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. 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 ${A}^{-1}B$. 19
• Test your understanding of Section 2.4. 35-54.
• Solve a system of linear equations by inverting the coeﬃcient matrix. 57.
• 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
9.session:
Topic: Determinants.
• 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
10.session:
Miniproject 2: (0-1) matrices, Kirchoﬀ’s laws
11.session:
Topic: Subspaces, basis for subspaces.
• 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 .
12.session:
Topic: Dimension, Rank and nullity.
• 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 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.
13.session:
Topic: Coordinatesystems.
• 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.
14.session:
Topic: Linear transformations and coordinate systems.
• 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 matrixrepresentations of linear transformations 20-23, 25-38
• Find ${\left[T\right]}_{\mathsc{ℬ}}$, the standardmatrix for $T$ and a rule for $T$ given $T\left({b}_{i}\right)$ for all $b\in \mathsc{ℬ}$. 47, 49, 51
• Find ${\left[T\right]}_{\mathsc{ℬ}}$ udfra $T\left({b}_{i}\right)$ as a linearcombination of $\mathsc{ℬ}$. Then ﬁnd $T\left(w\right)$, where $w$ is a linearcombination of $\mathsc{ℬ}$. 39, 55 43,59
15.session:
Topic: Eigenvectors og og eigenvalues. 5.1 and 5.2 till p. 307
• 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 ${B}^{T}$ 83.
16.session:
Topic: Diagonalization. 5.3
• Section 5.3
• Given a matrix $A$ and the characteristic polynomial. Find $P$ and a diagonalmatrix $D$, 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 unkonown. For which values is the matrix not diagonalizable. 63
• Section 5.5. These exercises are connected to miniproject 3.
• Find the general solution to a system of diﬀerential equations.. 45
17.session:
Miniproject 3: Systems of diﬀ. eq.’s, 5.5
18.session:
Topic: Ortogonality, Gram Schmidt, QR-faktorization.
• 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
• $QR$-faktorization. 25,27,29, 31
• Solve systems of equations using $QR$-faktorization. 33, 35, 37,39 OBS: Show that the solutions you found to $Rx={Q}^{T}b$ are solutions to $Ax=b$. (An extra challenge: Why is this necessary.)
• Test your understanding of Gram-Schmidt and $QR$-faktorization. 41-52
19.session:
Topic: Ortogonale projektioner. 6.3
• 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 og 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
20.session:
Topic: Orthogonal matrices. Orthogonal transformations in the plane. 6.5 till p. 419
• 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
• Rigid motions in the plane. Find $Q$ and $b$, s.t. $F\left(v\right)=Qv+b$ for all $v$. 61, 63
• 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.

21.session:
Miniproject 4: Least squares, 6.4
22.session:
Topic: Rigid motion. 6.5 p.419-421.

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.
• Section 6.5
• Determine the matrix and vector of a rigid motion. 61, 62, 63, 64
• Old exams.

## Miniprojects

### Miniprojekt 1

Miniproject 1 is supported by screencast 1, 2 and 3 that are available in the MATLAB center.

The webpage http://www.mathworks.com/help/techdoc/math/f4-983672.html in the PDF file is no longer available. Please use http://mathworks.com/help/matlab/math/systems-of-linear-equations.html instead.

Literature: Appendix D

### Miniprojekt 2

Miniproject 2 is supported by screencast 4 that is only available in Danish (Danish MATLAB center) at the moment.

Literature: Appendix D

### Miniprojekt 3

Click here to download the Matlab code mentioned in the miniproject.. Note, that it is a zip compressed file consisting of 5 files.

Miniproject 3 is supported by screencast 6 that is available in the MATLAB center.

Literature: Appendix D

### Miniprojekt 4

The following MATLAB files are used in the miniproject:

Miniproject 4 is supported by screencast 7 that is available in the MATLAB center.

Literature: Appendix D

## Old exams

Note: new structure in the organisation of the exam. Relevant from spring 2016 and onwards.
• 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
• Miniprojects 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 the new 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. The teaching assistants are preparing to help with the material from the last few lectures, and they might not be able to help with all your math questions, but feel free to ask. This is a new initiative and its success is partly measured by the amount of students coming to the math cafe. If there is a great interest in this initiative we will schedule more than the ones planed now. On the other hand if attendance is very low cancellation may occur.

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

Here the math cafe in general runs Tuesday afternoon every week.
Current scheduled dates (will be updated throughout the semester):
• Tuesday 11/10-16 16:15-17:45 in auditorium 2.
• Tuesday 18/10-16 16:15-17:45 in auditorium 2.
• Tuesday 25/10-16 16:15-17:45 in auditorium 2.
• Tuesday 1/11-16 16:15-17:45 in auditorium 2.
• Tuesday 8/11-16 16:15-17:45 in auditorium 2.
• Tuesday 15/11-16 16:15-17:45 in auditorium 2.
• Tuesday 22/11-16 16:15-17:45 in auditorium 2.
• Tuesday 29/11-16 16:15-17:45 in auditorium 2.
• Tuesday 6/12-16 16:15-17:45 in auditorium 2.

### Esbjerg

Here the math cafe in general runs Monday afternoon every other week (but we start with a Tuesday). Scheduled dates so far (will be updated throughout the semester):
• Tuesday 11/10-16 16:15-17:45 in room C1.117.
• Monday 24/10-16 16:15-17:45 in room C1.117.
• Monday 7/11-16 16:15-17:45 in room C1.117.
• Monday 21/11-16 16:15-17:45 in room C1.117.
• Monday 5/12-16 16:15-17:45 in room C1.117.

### Copenhagen

Here the math cafe in general runs Friday afternoon every other week.
Scheduled dates so far (will be updated throughout the semester):
• Friday 21/10-16 16:15-17:45 in room 0.108, FKJ 10A.
• Friday 4/11-16 16:15-17:45 in room 0.108, FKJ 10A.
• Friday 18/11-16 16:15-17:45 in room 0.108, FKJ 10A.
• Friday 2/12-16 16:15-17:45 in room 0.108, FKJ 10A.
• Friday 16/12-16 16:15-17:45 in room 0.108, FKJ 10A.

## Mathematics during the weekend

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

Then Math Saturday the 5th of November at 9:30-15:00 is just what you need. The main part of this event is held as a workshop in the canteen at Strandvejen 12-14, but we will also use Auditorium A.223 for a short introduction of the projects of the day.

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. The two projects will make use of e.g. matrix multiplication, rotation matrices, scaling and translation, and you will use a large part of the material you have learnt in the semester so far. Through both "pen and paper" exercises and MATLAB exercises the projects will stregthen your math skills. Hence, this is a great occacion 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. The two subjects are Image representations and Computer graphics and planetary orbits, and a more detailed description of each is available on the Danish version of this page (you may try your luck with Google Translate or ask a fellow student that understands Danish).

A free sandwich is served for lunch and therefore you need to sign up by filling out the form below no later than Wednesday the 2nd of November.

## Preparation for re-exam

You are offered help to prepare for the coming exam in both calculus and linear algebra at all three campuses. The idea is much like the exercise session during a normal lecture, where you on your own solve exercises and can get help from a teaching assistant present to help you. The point of departure for this exam preparation is the old exam sets available at this website, and you are encouraged to solve as much as possible on your own before showing up, so you know which parts you find difficult. Please note that the teaching assistants are not coming to the group rooms; rather everybody sits in the same room where the teaching assistants are present as indicated below.

### Aalborg - calculus

Saturday 11 February 12:30 - 16:30 in room 314 Strandvejen 12.

Sunday 12 February 12:30 - 16:30 in room 314 Strandvejen 12.

### Aalborg - Linear algebra

Saturday 11 February 12:30 - 16:30 in room 315 Strandvejen 12.

Sunday 12 February 12:30 - 16:30 in room 315 Strandvejen 12.

### Esbjerg - Calculus

Tuesday 14 February 14:30 - 16:15 in room C1.117.

### Copenhagen - Linear algebra

Saturday 11 February 10:00 - 12:00 in room 0.108 FKJ 10A.