Linear algebra

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.

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

2. session:

Topic: Matrix-vector product and systems of linear equations

Exercises:

• Section 1.2.
  • Write $2\times 2$ 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.

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

4. session:

Topic: Linear independence.

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.

5. session:

Topic: Linear transformations and matrices.

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?
    • $f:ℝ\to ℝ$, $f\left(x\right)=x^2+1$
    • $g:ℝ\to ℝ$, $g\left(x\right)=x^3+1$
    • $h:\text{The set of Danish citizens}\to ℝ$ $h\left(x\right)$ 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

6. session:

Topic: Matrix multiplication, composition of linear transformations.

Exercises:

• Section 2.1.
  • If the product of two matrices is defined, find the size, $m\times n$, 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

7. session:

Topic: Invertible matrices and invertible linear transformations.

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. 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

8. session:

Topic: Determinants.

Exercises:

• Section 3.1
  • Determinant of a $2\times 2$ matrix. 1, 3, 7. Do the calculation using the formula on p. 200.
  • Determinant of a $3\times 3$ matrix using cofactors. 13, 15
  • Calculate determinants - choose your preferred method. 21, 23.
  • Determinant of $2\times 2$ 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 $2\times 2$ matrices. 71
• Section 3.2 Prove that $det\left(B^{-1}AB\right)=det\left(A\right)$ for $n\times n$ matrices $A$ and $B$, where $B$ is invertible. 71

9. session:

Topic: Subspaces, basis for subspaces.

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.

10. session:

Topic: Dimension, Rank and nullity.

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 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.

11. session:

Topic: Coordinatesystems.

Exercises:

• Section 4.4.
  • Find $v$ given ${\left[v\right]}_{\mathcal{ℬ}}$ and $\mathcal{ℬ}$. 1, 7
  • Given $v$ as a linear combination of $\mathcal{ℬ}$, what is ${\left[v\right]}_{\mathcal{ℬ}}$? 13
  • Find ${\left[v\right]}_{\mathcal{ℬ}}$ given $\mathcal{ℬ}$ 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 $\left[{\left[e_1\right]}_{\mathcal{ℬ}}{\left[e_2\right]}_{\mathcal{ℬ}}\right]$ and the matrix whose columns are the vectors in $\mathcal{ℬ}$. 51, 53
  • A basis $\mathcal{ℬ}$ for the plane is constructed by rotating the standard basis. What is the connection between $v$ and ${\left[v\right]}_{\mathcal{ℬ}}$. 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.

12. session:

Topic: Linear transformations and coordinate systems.

Exercises:

• Section 4.5
  • Find the matrix for $T$ wrt. $\mathcal{ℬ}$. 1,3,7
  • Find the standard matrix for $T$ given ${\left[T\right]}_{\mathcal{ℬ}}$ and $\mathcal{ℬ}$. 11, 15
  • Test your understanding of matrixrepresentations of linear transformations 20-23, 25-38
  • Find ${\left[T\right]}_{\mathcal{ℬ}}$, the standardmatrix for $T$ and a rule for $T$ given $T\left(b_i\right)$ for all $b\in \mathcal{ℬ}$. 47, 49, 51
  • Find ${\left[T\right]}_{\mathcal{ℬ}}$ udfra $T\left(b_i\right)$ as a linearcombination of $\mathcal{ℬ}$. Then find $T\left(w\right)$, where $w$ is a linearcombination of $\mathcal{ℬ}$. 39, 55 43,59

13. session:

Topic: Eigenvectors og og eigenvalues. 5.1 and 5.2 till 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 $2\times 2$ 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.

14.session:

Topic: Diagonalization. 5.3

Exercises:

• 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 self-study session 3.
  • Find the general solution to a system of differential equations.. 45

15. session:

Topic: Ortogonality, Gram Schmidt, QR-faktorization.

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: $y_1\left(t\right)=-e^{-3t}+2e^{4t}$. $y_2\left(t\right)=3e^{-3t}+e^{4t}$)
• 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

16. session:

Topic: Ortogonale projektioner. 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 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

17. session:

Topic: Orthogonal matrices. Orthogonal transformations in the plane. 6.5 till p. 419

Exercises:

• Section 6.5
  • Recognize an orthogonal matrix. 1,4,5,3
  • Decide, if an orthogonal $2\times 2$ 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 $Q_x$ and $Q_z$ be the matrices for a $90^{\circ }$ rotation around the $x$-axis and the $z$-axis respectively. $$Q_x=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]Q_z=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$$

    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(${\left[1-11\right]}^T$))

• Part II of the exam January 2013

  Notice the different ways of posing multiple choice problems.

18. 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.

Exercises:

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

Self-study sessions

Self-study session 1

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

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:

• polynomial_data.txt
• linreg.m
• linreg2.m
• example.m

Self-study session 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.

• 2018 spring
  • Regular exam (answers)
  • Reexam (answers)
• 2017 autumn
  • Regular exam (answers)
  • Reexam (answers)
• 2017 spring
  • Regular exam (answers)
  • Reexam (answers)
• 2016 autumn
  • Regular exam (answers)
  • Reexam (answers)
• 2016 spring
  • Regular exam (answers)
  • Reexam (answers)
• Test set
  • Test set 2016-01 (answers where some parts may be in Danish)

Previous exams

• Test set (2015 autumn)
  • Test set 2015-01 (answers where some parts may be in Danish)
• Test sets
  • Test set 1 (answers where some parts may be in Danish)
  • Test set 2
• 2010
  • Regular exam (answers where some parts may be in Danish)
  • Re exam (answers where some parts may be in Danish)
• 2011
  • Regular exam (answers where some parts may be in Danish)
  • Re exam (answers where some parts may be in Danish)
• 2012
  • Regular exam (answers where some parts may be in Danish)
  • Re exam (answers where some parts may be in Danish)
• 2013
  • Regular exam (including answers where some parts may be in Danish)
  • Re exam (including answers where some parts may be in Danish)
• 2014
  • Regular exam (answers where some parts may be in Danish)
  • Re exam (including answers where some parts may be in Danish)
• 2015 autumn
  • Regular exam (answers where some parts may be in Danish)
  • Re exam (answers where some parts may be in Danish)

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.

Information on when and where the math cafe will take place is coming soon.

Aalborg (email: mfindi13@student.aau.dk)

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

• Tuesday 6/3-18 16:15-17:45 in Auditorium 2.
• Thursday 8/3-18 16:15-17:45 in Auditorium 2.
• Tuesday 13/3-18 16:15-17:45 in Auditorium 2.
• Tuesday 20/3-18 16:15-17:45 in Auditorium 2.
• Thursday 22/3-18 16:15-17:45 in Auditorium 2.
• Tuesday 27/3-18 16:15-17:45 in Auditorium 2.
• Tuesday 3/4-18 16:15-17:45 in Auditorium 2.
• Thursday 5/4-18 16:15-17:45 in Auditorium 2.
• Tuesday 10/4-18 16:15-17:45 in Auditorium 2.
• Tuesday 17/4-18 16:15-17:45 in Auditorium 2.
• Thursday 19/4-18 16:15-17:45 in Auditorium 2.
• Tuesday 24/4-18 16:15-17:45 in Auditorium 2.
• Tuesday 1/5-18 16:15-17:45 in Auditorium 2.
• Thursday 3/5-18 16:15-17:45 in Auditorium 2.
• Tuesday 8/5-18 16:15-17:45 in Auditorium 2.
• Tuesday 15/5-18 16:15-17:45 in Auditorium 2.
• Thursday 17/5-18 16:15-17:45 in Auditorium 2.
• Tuesday 22/5-18 16:15-17:45 in Auditorium 2.
• Tuesday 29/5-18 16:15-17:45 in Auditorium 2.
• Thursday 31/5-18 16:15-17:45 in Auditorium 2.

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 14/3-18 16:15-17:45 in room B202.
• Wednesday 21/3-18 16:15-17:45 in room B202.
• Wednesday 28/3-18 16:15-17:45 in room B202.
• Wednesday 4/4-18 16:15-17:45 in room B202.
• Wednesday 11/4-18 16:15-17:45 in room B202.
• Wednesday 18/4-18 16:15-17:45 in room B202.
• Wednesday 25/4-18 16:15-17:45 in room B202.
• Wednesday 16/5-18 16:15-17:45 in room B202.
• Wednesday 23/5-18 16:15-17:45 in room B202.
• Wednesday 30/5-18 16:15-17:45 in room B202.

Copenhagen (email: ahmad.zafari@ymail.com)

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

• Tuesday 20/3-18 16:15-17:45 in room 0.108, Building D.
• Tuesday 27/3-18 16:15-17:45 in room 0.108, Building D.
• Tuesday 3/4-18 16:15-17:45 in room 0.108, Building D.
• Tuesday 10/4-18 16:15-17:45 in room 0.108, Building D.
• Tuesday 17/4-18 16:15-17:45 in room 0.108, Building D.
• Tuesday 24/4-18 16:15-17:45 in room 0.108, Building D.
• Tuesday 1/5-18 16:15-17:45 in room 0.108, Building D.
• Tuesday 8/5-18 16:15-17:45 in room 0.108, Building D.
• Tuesday 15/5-18 16:15-17:45 in room 0.108, Building D.
• Friday 25/5-18 16:15-17:45 in room 0.108, Building D.

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. The concept consists of two parts. First, a teacher will solve a number of exercises on the blackboard. Afterwards, there will 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.

We urge you to participate from the beginning in order not to disturb during the first part of the session.

Aalborg

Four teaching assistants will be available to help you while you prepare for the exam. They are present in AUD 6 and 7 on Tuesday the 12^th of June at 13:00-16:00 and Wednesday the 13^th of June at 12:00-17:00.

Before the re-exam there will be a Q&A-session on Tuesday the 22^nd of August and Wednesday the 23^rd of August. This takes place in AUD 2 at 9:00–12:00.

Esbjerg

There are Q&A-sessions Wednesday the 13^th of June and Thursday the 14^th of June, both days at 9:00-10:30. This takes place in C1.119.

Before the re-exam there will be a Q&A-session on Tuesday the 22^nd of August at 10:00–12:00. This takes place in C.1.119.