Linear algebra

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 $2\times 2$ rotation matrices. 17

3.session:

Gauss-elimination. Span.

• 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

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 $v$ 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

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 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.

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 $T:{ℝ}^n\to {ℝ}^m$. 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

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. 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

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. 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

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 $2\times 2$ 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

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 $A$ 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.

Type 2: Section 4.4, exercise 1, 13.

14.session:

Linear transformations and coordinate systems.

• Section 4.4.
  • Find $v$ given ${\left[v\right]}_{\mathcal{ℬ}}$ 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.

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. $\mathcal{ℬ}$. 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

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 $2\times 2$ 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.

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 $A$ 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
  • In exercise 45, find the solution satisfying $y_1\left(0\right)=1$ og $y_2\left(0\right)=4$.(Solution: $y_1\left(t\right)=-e^{-3t}+2e^{4t}$. $y_2\left(t\right)=3e^{-3t}+e^{4t}$)

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
  • $QR$-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

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. 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

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.

• 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
  • 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 $90^{\circ }$ 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
  http://www.first.math.aau.dk/digitalAssets/62/62212_lial_8jan_13_english.pdf

  Notice the different ways of posing multiple choice problems.

Miniprojects

Miniprojekt 1

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

Miniprojekt 2

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

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.

Miniprojekt 4

The following MATLAB files are used in the miniproject:

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

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

Old exams

• 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)

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}