Literature
- [SIF] L. E. Spence, A. J. Insel, og S. H. Friedberg, "Elementary Linear Algebra: A Matrix Approach," 2nd Edition, Pearson, Prentice Hall, 2008.
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. One should ACTIVELY participate in the four mini-projects in order to be allowed to take the written exam. It is considered that a student has actively participated if the teacher registers that the student has participated in at least three out of the four mini-projects. If a student only participates in two mini-projects, then the remaining two mini-projects should be completed individually and submitted to the teacher for evaluation. If a student only participates in one mini-project, then the remaining three mini-projects should be completed individually and submitted to the teacher for evaluation. If a student has not participated in any mini-project, then the four mini-projects should be completed individually and submitted to the teacher for evaluation.
.
Book guide
The book for this course is
- [SIF] L. E. Spence, A. J. Insel, og S. H. Friedberg, "Elementary Linear Algebra: A Matrix Approach," 2nd Edition, Pearson, Prentice Hall, 2008.
Unfortunately, the publisher has made some mistakes, which mean that three versions of the book exist. We are of course very sorry for this inconvenience. Below, we try to explain where the material covered in this course is located in the different versions of the books. Please, see the plan at the bottom of this page.
Chapter |
Book 1: Original book |
Book 2: Book sold last year and this year |
Book 3: Newest book |
|
Page numbers at the top ISBN: 0-13-158034-5 |
Page numbers both at the top and bottom ISBN: 978-1-292-02503-2 |
Page numbers at the bottom ISBN: 978-1-292-02503-2 |
Chapter 1-5 |
Chapter 1-5 are the same for all three books |
"Orthogonality" |
Chapter 6 p. 359-486 |
Chapter 6 p. 359-486 p. 423-550 |
Chapter 7 p. 423-550 (Note that this is chapter 6 in the answers |
"Vector spaces" |
Chapter 7 p. 489-549 |
Chapter 7 p. 489-549 p. 353-421 |
Chapter 6 p. 353-421 (Note that this is chapter 7 in the answers) |
Plan
-
1.session:
- Introduction to vectors and matrices: 1.1, 6.1 p.
361-366, on pp. 364 - 365 only the theorems. (Book 3/newest book: 7.1. p. 425-430, on pp. 428 - 429 only the theorems.) 1.2 till p.19 .
-
2.session:
- Matrixvectorproduct systems of linear equations: 1.2
from p. 19, 1.3
-
3.session:
- Gauss-elimination. Span. 1.4 og 1.6
-
4.session:
- Linear independence 1.7
-
5.session:
- Miniproject 1.(Solve systems of linear equations using
MatLab)
-
6.session:
- Linear transformations and matrices.. 2.7. 2.8 til s. 185
mid. (In general about functions (Injective, surjective bijective),
Appendix B)
-
7.session:
- Matrix multiplication, composition of linear
transformations. 2.1 og 2.8 p.185 mid, till 187
-
8.session:
- Invertible matrices and invertible linear
transformations. 2.3, 2.4 og 2.8 p.187-188
-
9.session:
- Determinants. 3.1 og 3.2 till p. 217 l.9.
-
10.session:
- Miniproject 2 (0-1 matrices, Kirchoff’s law)
-
11.session:
- Subspaces, basis for subspaces. 4.1 og 4.2 till p.245,
mid.
-
12.session:
- Dimension, Rank and nullity. the remaining part of
4.2, 4.3
-
13.session:
- Coordinate systems. 4.4
-
14.session:
- Linear transformations and coordinate systems. 4.5
-
15.session:
- Eigenvectors and og eigenvalues. 5.1 og 5.2 till p. 307
-
16.session:
- Diagonalization. 5.3
-
17.session:
- Miniproject 3 (Systems of differential equations, 5.5)
-
18.session:
- Orthogonality, Gram Schmidt, QR-faktorisation. 6.2 (Book 3/newest book: 7.2)
-
19.session:
- Orthogonal projection. 6.3. (Book 3/newest book: 7.3)
-
20.session:
- Orthogonal matrices Ortogonal transformations in the
plane. 6.5 til s. 419. (Book 3/newest book: 7.5 til p. 483)
-
21.session:
- Miniproject 4 (Method of least squares, 6.4 (Book 3/newest book: 7.4)
-
22.session:
- Rigid motion. 6.5: p.419-421 (Book 3/newest book: 7.5: p.483-485)
Problems
The exercises support two ways of structuring the sessions:
Type 1: Recap from last session, Exercises, Lecture 1,, Lecture
2. Type 2: Recap from last session, Lecture 1, Exercises, Lecture
2
The first and last session are structured with exercises in the last
two-hour slot.
Manual for the exercises:
- Exercises are structured according to content.
- Some are underlined. Do all the ones which are not underlined.
Then go back and do the ones which are underlined.
- 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, htere 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:
- Introduction to vectors and matrices.
Structure: Introduction and lecture. 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 1.1 Symmetric matrices 71, 72, 75.
-
2.session:
- Matrix-vector product and systems of linear equations:
- 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
- Skew matrices 79, 80, 81
For type 2 sessions: Section 1.2. Write
rotation matrices. 17
-
3.session:
- Gauss-elimination. Span.
- Section 1.2.
- Write
rotation matrices. 17, 19
- Test your understanding of matrix-vector products.
51-64
- Section 1.3.
- Write the coefficient matrix and the augmented
matrix of a linear system: 1,3,5.
- 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
Type 2: Section 1.4. Decide, if a linear system is consisten. If so, find
the general solution. 3
-
4.session:
- Linear independence.
- 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
in Span( )?.
1,3,7
- Is
in Span()?
A coordinate in
is unknown. 17, 19
- Is
consistent for all ?
31,33.
- Test your understanding of span. 45-64.
- About the connection between Span()
and the span of a linearcombination of .
71, 72. Consequences for row-operations: 77, 78.
- Section 1.4:
- Systems of equations where a coefficient
is unknown. For which values of
is the system inconsistent. 17, 19,21
Type 2: Section 1.7 exercises 1, 5.
-
5.session:
- Miniproject 1.(Solve systems of linear equations using
MatLab)
-
6.session:
- Linear transformations and matrices.
- Section 1.7.
- Determine, if a set of vectors is linearly dependent.
1,5,7,9,11
- Find a small subset of ,
with the same span as .13,
15.
- Determine, if a set of vectors is linearly independent.
23,25,27
- Test your understanding of linear (in)dependence 1.7
63-82.
- Given a set of vectors, one of which has an unknown
coordinate .
For which values of ,
if any, is the set linearly dependent. 41.
Type 2. Section 2.7 Exercise 1, 3
-
7.session:
- Matrix multiplication, composition of linear transformations.
- Section 2.7.
-
is induced by a matirx. Find
and .
1, 3
- Find the image of a vector under a linear
transformation induced by a matrix. 7, 11
- From the rule for ,
find
and ,
such that .
21 23
- Find the standard matrix of a linear transformation.
25, 27, 29,31, 33
- Test your understanding of linear transformations
and their matrix representations. 35-54.
- Section 2.8.
- Find a generating set for the range. 1,3
- Are the following maps surjective (onto), injective
(one-to-one), bijective?
- ,
- ,
-
is the CPR-number for .
- 61, 65.
- Determine by finding a spanning set of the null
space, 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
er linear and
is known, what is .
57
- Determine, if
is linear. 77, 73, 79
Type 2: Section 2.1 exercise 1 og 5.
-
8.session:
- Invertible matrices and invertible linear transformations.
- Section 2.1.
- If the product of two matrices is defined, find the size,
,
of the product. 1,3
- Calculate matrix products. 5,9,11,7. Calculate a
given entrance in a product matrix. 25
- Test your understanding of the matrix product.
33-50.
- Section 2.8.
- Find a rule for
from rules for
og .
69. Find standard matrices for ,
og .
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
Type 2: Section 2.3 exercises 1, 3.
-
9.session:
- Determinants.
- Section 2.3.
- determine whether .
1,3
- Given
and .
Find the inverse of combinations of
and .
9, 11.
- Elementary matrices. Find inverses. 17, 19. Givet
,
,
find elementary matrices ,
such that .
25, 29.
- Section 2.4. Is a given matrix invertible? If so, find the inverse.
1, 3, 5, 9, 13
- Section 2.8 The connection between invertible matrices and
invertible linear transformations. 59,60.
- Section 2.4.
- Rowreduction to calculate .
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
of
and a
s.t. .
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
Type 2: Section 3.1 opgave 1, 3.
-
10.session:
- Miniproject 2 (0-1 matrices, Kirchoff’s laws)
-
11.session:
- Subspaces, basis for subspaces.
- Section 3.1
- Determinant of a
matrix. 1, 3, 7. Do the calculation using the formula
on p. 200.
- Determinant of a
matrix using cofactors. 13, 15
- Calculate determinants - choose your preferred
method. 21, 23.
- Determinant of
matrices and area. 29
- Determinant and invertibility. 37.
- Test your understanding of determinants and
cofactors. 45-64
- Section 3.2
- Calculate determinants- develop after a given column
1, 5
- Calculate determinants using row-operations . 13, 15,
21, 23
- Test your understanding of the properties of
determinants. 39-58.
- Section 3.1 Prove that
for
matrices. 71
- Section 3.2 Prove that
for
matrices
and ,
where
is invertible. 71
Type 2: Section 4.1 Exercises 1, 11.
-
12.session:
- Dimension, Rank and nullity.
- 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
.
Type 2: Section 4.2 opgave 17
-
13.session:
- Coordinatesystems.
- Section 4.2
- Find a basis for the range and null space of a linear
transformation. 9, 11, 13 15
- Find a basis for a subspace 17, 19, 23
- Test your understanding of Basis and dimension.
33-52.
- Section 4.3.
- Find the dimension of the column space,
null space and row space of a matrix
and the null space of
- When
is on reduced echelon form. 1, 3.
- In general. 7.
- Find the dimension of a subspace. 15
- Find en basis for 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.
Type 2: Section 4.4, exercise 1, 13.
-
14.session:
- Linear transformations and coordinate systems.
- Section 4.4.
- Find
given
and .
1, 7
- Given
as a linear combination of ,
what is ?
13
- Find
given
and .
15, 17, 19
- Write a vector as a linear combination of a set of
vectors. 25, 27
- Test your understanding of coordinate systems. 31-50
- What is the connection between the matrix
and the matrix whose columns are the vectors in
.
51, 53
- A basis
for the plane is constructed by rotating the standard
basis. What is the connection between
and .
55, 67, 75
- Equations for cone sections before and after change
of basis. 79
- What does it imply, that there is a vector ,
s.t. ?
99.
Type 2: Section 4.5 opgave 1, 3.
-
15.session:
- Eigenvectors og og eigenvalues. 5.1 and 5.2 till p.
307
- Section 4.5
- Find the matrix for
wrt. .
1,3,7
- Find the standard matrix for
given
and .
11, 15
- Test your understanding of matrixrepresentations of
linear transformations 20-23, 25-38
- Find ,
the standardmatrix for
and a rule for
given
for all .
47, 49, 51
- Find
udfra
as a linearcombination of .
Then find ,
where
is a linearcombination of .
39, 55 43,59
Type 2: Section 5.1 opg. 3, 7.
-
16.session:
- Diagonalization. 5.3
- Section 5.1
- Show that a vector is an eigenvector. 3, 7
- Show that a scalar is an eigenvalue. 13, 21
- Test your understanding of eigenvalues and
eigenvectors. 41-56, 57-60
- Section 5.2
- Find eigenvalues and a basis for the associated
eigenspaces
- For a matrix - given the characteristic
polynomial 1, 11
- For a matrix. 15, 19
- For a linear transformation - given the
characteristic polynomial. 31
- For a linear transformation. 37
- Does a
matrix have any (reat) eigenvalues? 41
- Test your understanding of characteristic polynomial,
multiplicity of eigenvalues. 53-59, 61,63-65, 69-72.
- Connection between eigenspaces for
og
81.
- Connection between eigenvalues (and egenvectors?) for
og
83.
Type 2: Section 5.3 opg. 1 og 3.
-
17.session:
- Miniproject 3 (Systems of diff. eq.’s, 5.5)
-
18.session:
- Ortogonality, Gram Schmidt, QR-faktorization.
- Section 5.5. These exercises are connected to miniproject
3.
- Test your understanding of systems of linear
differential equations. 8-11
- Section 5.3
- Given a matrix
and the characteristic polynomial. Find
and a diagonalmatrix ,
s.t.
or explain why
is not diagonalizable. 1, 3, 5,7,9
- As above, but the characteristic polynomial is not
given. 13, 15 17
- Test your understanding of diagonalization of
matrices. 29-37, 39-43, 45,46
- Determine from the eigenvalues and their multiplicity
whether
is diagonalizable. 49, 51
- Given eigenvalues and a basis for the eigenspaces,
find .
57, 59
- Given a matrix and the characteristic polynomial.
One entrance is an 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
- In exercise 45, find the solution satisfying
og .(Solution:
.
)
Type 2: Section 6.2 opg. 1,3.
-
19.session:
- Ortogonale projektioner. 6.3
- 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
- -faktorization.
25,27,29, 31
- Solve systems of equations using -faktorization.
33, 35, 37,39 OBS: Show that the solutions you found
to
are solutions to .
(An extra challenge: Why is this necessary.)
- Test your understanding of Gram-Schmidt and -faktorization.
41-52
Type 2: Section 6.3 opg. 1, 3
-
20.session:
- Orthogonal matrices. Orthogonal transformations in the
plane. 6.5 till p. 419
- Section 6.1 (refresh your memory) Projection on a line. 43,
45
- Section 6.3
- Find a basis for the orthogonal complement. 1, 3, 5
- write a vector
as a sum ,
where
and .
9,11
- As above. Moreover, find the matrix
for orthogonal projection on ,
find the distance to .
17,19,21 Hint to 21: Warning - the columns of
are not linearly independent.
- Test your understanding of orthogonal projection og
orthogonal complement. 33-56.
- What is the orthogonal complement to the
orthogonal complement? 63
- What is
and .
67
- Find
given an orthonormal basis for .
75
Type 2: Section 6.5 opg. 1, 4.
-
21.session:
- Miniproject 4 (Least squares, 6.4)
-
22.session:
- 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.
The structure of this session is Lectures first, exercises last.
Old exams
- Test sets
- 2010
- 2011
- 2012
- 2013
- 2014
Book guide
The book for this course is
- [SIF] L. E. Spence, A. J. Insel, og S. H. Friedberg, "Elementary Linear Algebra: A Matrix Approach," 2nd Edition, Pearson, Prentice Hall, 2008.
Unfortunately, the publisher has made some mistakes, which mean that three versions of the book exist. We are of course very sorry for this inconvenience. Below, we try to explain where the material covered in this course is located in the different versions of the books. Please, see the curriculum at the bottom of this page.
Chapter |
Book 1: Original book |
Book 2: Book sold last year and this year |
Book 3: Newest book |
|
Page numbers at the top ISBN: 0-13-158034-5 |
Page numbers both at the top and bottom ISBN: 978-1-292-02503-2 |
Page numbers at the bottom ISBN: 978-1-292-02503-2 |
Chapter 1-5 |
Chapter 1-5 are the same for all three books |
"Orthogonality" |
Chapter 6 p. 359-486 |
Chapter 6 p. 359-486 p. 423-550 |
Chapter 7 p. 423-550 (Note that this is chapter 6 in the answers |
"Vector spaces" |
Chapter 7 p. 489-549 |
Chapter 7 p. 489-549 p. 353-421 |
Chapter 6 p. 353-421 (Note that this is chapter 7 in the answers) |
Curriculum
[SIF] L. E. Spence, A. J. Insel, og S. H. Friedberg, "Elementary Linear Algebra: A Matrix Approach," 2nd Edition, Pearson, Prentice Hall, 2008:
- 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: Book 1 and 2: Section 6.1 to page 366, 6.2, 6.3, 6.5.. Book 3: Section 7.1 to page 430, 7.2, 7.3, 7.5