Preparation for reexam
Soon some you unfortunately have to take the exam again, and you might wonder (again):
How do I prepare for the reexam in February in the best way possible?
The answer is:
Read the book and do exercises (mainly the old exams). Note that 'Testset 1' is closest to the new structure of the exam (with multiple choice exercises). Afterwards, do more exercises (old exams) and finally: do even more of the exercises (possibly old exam questions)!
Aalborg
To support the exam preparation a math bootcamp where teaching assistants are available will be held during the weekend before the reexam: Sat 13 Feb and Sun 14 Feb at 12:3016:30 both days.
This happens in room A413 and A414 at Strandvejen 1214, Aalborg, which is reserved for this purpose. Hence, during the above mentioned times, you can sit (individuellly or in small groups) and do exercises and teaching assistants will be present to help you. Focus is on the problems from the ordinary exam and "testset 1". Questions not related to these might not be answered.
IMPORTANT: Teaching assistants do not supply complete answers to the exercises, but help with specific questions (for improving your learning). Therefore it will be an advantage to prepare from home by doing exercises.
København
To support the exam preparation an extra math cafe is planned where a teaching assistant is present Friday 12 February at 14:0016:00 in room 0.108, FKJ 10A.
At this math cafe, you can sit (individuellly or in small groups) and do exercises and a teaching assistant will be present to help you. Focus is on the problems from the ordinary exam and "testset 1". Questions not related to these might not be answered.
IMPORTANT: Teaching assistants do not supply complete answers to the exercises, but help with specific questions (for improving your learning). Therefore it will be an advantage to prepare from home by doing exercises.
Literature
 [SIF] L. E. Spence, A. J. Insel, og S. H. Friedberg, "Elementary Linear Algebra: A Matrix Approach," 2nd Edition, Pearson, Prentice Hall, 2008. This book is identical to Compiled by Olav Geil, "Elementary Linear Algebra," Pearson, 2015.
Supplementary literature
MATLAB
The use of Matlab is an integral part of the four sessions without lectures (miniprojects) 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 miniprojects 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 miniprojects. If a student only participates in two miniprojects, then the remaining two miniprojects should be completed individually and submitted to the teacher for evaluation. If a student only participates in one miniproject, then the remaining three miniprojects should be completed individually and submitted to the teacher for evaluation. If a student has not participated in any miniproject, then the four miniprojects should be completed individually and submitted to the teacher for evaluation.
.
Also look at Moodle.
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 4: Compiled by Olav Geil, "Elementary Linear Algebra," Pearson, 2015 
Book 2: Special print 1 
Book 3: Special print 2 
Book 4: Compiled by Olav Geil, "Elementary Linear Algebra," Pearson, 2015 

Page numbers at the top ISBN: 0131580345 
Page numbers both at the top and bottom ISBN: 9781292025032 
Page numbers at the bottom ISBN: 9781292025032 
Page numbers at the top ISBN: 9781784483722 
Chapter 15 
Chapter 15 are the same for all three books 
"Orthogonality" 
Chapter 6 p. 359486 
Chapter 6 p. 359486 p. 423550 
Chapter 7 p. 423550 (Note that this is chapter 6 in the answers 
Chapter 6 p. 359486 
"Vector spaces" 
Chapter 7 p. 489549 
Chapter 7 p. 489549 p. 353421 
Chapter 6 p. 353421 (Note that this is chapter 7 in the answers) 
Chapter 7 p. 489549 
Plan

1.session:
 Introduction to vectors and matrices: 1.1, 6.1 p.
361366, on pp. 364  365 only the theorems. (Book 3: 7.1. p. 425430, 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:
 Gausselimination. 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.187188

9.session:
 Determinants. 3.1 og 3.2 till p. 217 l.9.

10.session:
 Miniproject 2 (01 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, QRfaktorisation. 6.2 (Book 3: 7.2)

19.session:
 Orthogonal projection. 6.3. (Book 3: 7.3)

20.session:
 Orthogonal matrices Ortogonal transformations in the
plane. 6.5 til s. 419. (Book 3: 7.5 til p. 483)

21.session:
 Miniproject 4 (Method of least squares, 6.4 (Book 3: 7.4)

22.session:
 Rigid motion. 6.5: p.419421 (Book 3: 7.5: p.483485)
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
twohour 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:
3739, 41,42, 4456.
 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:
 Matrixvector product and systems of linear equations:
 Section 1.2
 Matrixvector 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.
4551.
 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
$2\times 2$
rotation matrices. 17

3.session:
 Gausselimination. Span.
 Section 1.2.
 Write $2\times 2$
rotation matrices. 17, 19
 Test your understanding of matrixvector products.
5164
 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. 5776
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 Gausselimination: 5372.
 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. 4564.
 About the connection between Span($S$)
and the span of a linearcombination of $S$.
71, 72. Consequences for rowoperations: 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
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 $S$,
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
6382.
 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.
Type 2. Section 2.7 Exercise 1, 3

7.session:
 Matrix multiplication, composition of linear transformations.
 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 $T:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$.
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. 3554.
 Section 2.8.
 Find a generating set for the range. 1,3
 Are the following maps surjective (onto), injective
(onetoone), bijective?
 $f:\mathbb{R}\to \mathbb{R}$,
$f\left(x\right)={x}^{2}+1$
 $g:\mathbb{R}\to \mathbb{R}$,
$g\left(x\right)={x}^{3}+1$
 $h:\text{ThesetofDanishcitizens}\phantom{\rule{3.26288pt}{0ex}}\to \mathbb{R}$
$h\left(x\right)$
is the CPRnumber 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).
4155.
 Section 2.7.
 If $T$
er linear and $T\left(v\right)$
is known, what is $T\left(cv\right)$.
57
 Determine, if $T:{\mathbb{R}}^{n}\to {\mathbb{R}}^{m}$
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,
$m\times n$,
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.
3350.
 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. 5658.
 MatLab: Section 2.1 opg. 53
Type 2: Section 2.3 exercises 1, 3.

9.session:
 Determinants.
 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. 3554.
 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
Type 2: Section 3.1 opgave 1, 3.

10.session:
 Miniproject 2 (01 matrices, Kirchoﬀ’s laws)

11.session:
 Subspaces, basis for subspaces.
 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. 4564
 Section 3.2
 Calculate determinants develop after a given column
1, 5
 Calculate determinants using rowoperations . 13, 15,
21, 23
 Test your understanding of the properties of
determinants. 3958.
 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
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. 4362.
 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.
3352.
 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. 4160.
 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 $v$
given ${\left[v\right]}_{\mathcal{\mathcal{B}}}$
and $\mathcal{\mathcal{B}}$.
1, 7
 Given $v$
as a linear combination of $\mathcal{\mathcal{B}}$,
what is ${\left[v\right]}_{\mathcal{\mathcal{B}}}$?
13
 Find ${\left[v\right]}_{\mathcal{\mathcal{B}}}$
given $\mathcal{\mathcal{B}}$
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. 3150
 What is the connection between the matrix $\left[{\left[{e}_{1}\right]}_{\mathcal{\mathcal{B}}}{\left[{e}_{2}\right]}_{\mathcal{\mathcal{B}}}\right]$
and the matrix whose columns are the vectors in
$\mathcal{\mathcal{B}}$.
51, 53
 A basis $\mathcal{\mathcal{B}}$
for the plane is constructed by rotating the standard
basis. What is the connection between $v$
and ${\left[v\right]}_{\mathcal{\mathcal{B}}}$.
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.
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 $T$
wrt. $\mathcal{\mathcal{B}}$.
1,3,7
 Find the standard matrix for $T$
given ${\left[T\right]}_{\mathcal{\mathcal{B}}}$
and $\mathcal{\mathcal{B}}$.
11, 15
 Test your understanding of matrixrepresentations of
linear transformations 2023, 2538
 Find ${\left[T\right]}_{\mathcal{\mathcal{B}}}$,
the standardmatrix for $T$
and a rule for $T$
given $T\left({b}_{i}\right)$
for all $b\in \mathcal{\mathcal{B}}$.
47, 49, 51
 Find ${\left[T\right]}_{\mathcal{\mathcal{B}}}$
udfra $T\left({b}_{i}\right)$
as a linearcombination of $\mathcal{\mathcal{B}}$.
Then ﬁnd $T\left(w\right)$,
where $w$
is a linearcombination of $\mathcal{\mathcal{B}}$.
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. 4156, 5760
 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. 5359, 61,6365, 6972.
 Connection between eigenspaces for
$B$ og
$cB$
81.
 Connection between eigenvalues (and egenvectors?) for
$B$ og
${B}^{T}$
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, QRfaktorization.
 Section 5.5. These exercises are connected to miniproject
3.
 Test your understanding of systems of linear
diﬀerential equations. 811
 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. 2937, 3943, 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
 In exercise 45, ﬁnd the solution satisfying ${y}_{1}\left(0\right)=1$
og ${y}_{2}\left(0\right)=4$.(Solution:
${y}_{1}\left(t\right)={e}^{3t}+2{e}^{4t}$.
${y}_{2}\left(t\right)=3{e}^{3t}+{e}^{4t}$)
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. 6170, 7380
 Section 6.2
 Determine whether a set of vectors is orthogonal. 1,
3, 7
 Apply GramSchmidt. 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 GramSchmidt and $QR$faktorization.
4152
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 $u$
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. 3356.
 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
Type 2: Section 6.5 opg. 1, 4.

21.session:
 Miniproject 4 (Least squares, 6.4)

22.session:
 Rigid motion. 6.5 p.419421. 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
Note: new structure in the organisation of the exam. Relevant from autumn 2015 and onwards.
Previous exams
 Test sets
 2010
 2011
 2012
 2013
 2014
 2015 autumn
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: 0131580345 
Page numbers both at the top and bottom ISBN: 9781292025032 
Page numbers at the bottom ISBN: 9781292025032 
Chapter 15 
Chapter 15 are the same for all three books 
"Orthogonality" 
Chapter 6 p. 359486 
Chapter 6 p. 359486 p. 423550 
Chapter 7 p. 423550 (Note that this is chapter 6 in the answers 
"Vector spaces" 
Chapter 7 p. 489549 
Chapter 7 p. 489549 p. 353421 
Chapter 6 p. 353421 (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}