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Teaching Guide

CHAPTER 1 VECTORS

1.1 Vectors and Matrices in Engineering and Mathematics ; n-Space Scalars and Vectors • Equivalent Vectors • Vector Addition • Vector Subtraction • Scalar Multiplication • Vectors in Coordinate Systems • Components of a Vector Whose Initial Point Is Not at the Origin • Vectors in Rn • Equality of Vectors • Sums of Three or More Vectors • Parallel and Collinear Vectors • Linear Combinations • Application to Computer Color Models • Alternative Notations for Vectors • Matrices

Comments on 1.1: The material on geometric vectors in 2-space and 3-space is intended as a review and should be covered quickly; the primary focus should be on vectors in Rn. The commentary in "Looking Ahead" (p.9) may be worth emphasizing in class. Concept problems (such as that on p.8) occur throughout the text and can be assigned for classroom discussion, if desired.

1.2 Dot Product and Orthogonality; Norm of a Vector Unit Vectors • The Standard Unit Vectors • Distance Between Points in Rn • Dot Products • Algebraic Properties of the Dot Product • Angle Between Vectors in R2 and R3 • Orthogonality • Orthonormal Sets • Euclidean Geometry in Rn

Comments on 1.2: Some texts do not cover orthogonality until late in the course. We disagree with that approach. In keeping with our emphasis on developing the student's geometric intuition about the geometry of Rn, we cover this topic early. In this section the student will see how geometric results, such as the Theorem of Pythagoras in R2 and R3, generalize to Rn.

1.3 Vector Equations of Lines and Planes Vector and Parametric Equations of Lines • Lines Through Two Points • Point-Normal Equations of Planes • Vector and Parametric Equations of Planes • Lines and Planes in Rn • Comments on Terminology

Comments on 1.3: The material in this section, which will be new to some students and familiar to others, will be used in the next chapter to develop the geometric properties of solution sets of linear systems. This is why we have chosen to cover vectors before linear systems.

CHAPTER 2 SYSTEMS OF LINEAR EQUATIONS

2.1 Introduction to Systems of Linear Equations Linear Systems • Linear Systems with Two and Three Unknowns • Augmented Matrices and Elementary Row Operations

Comments on 2.1: Algorithms for solving linear systems will be discussed in the next section. Here we focus on terminology, matrix notation, and the idea of solving a linear system by applying elementary row operations to the augmented matrix. Theorem 2.1.1 is a key result. Example 7, which relates linear combinations and linear systems, will also prove to be important later, though it may not appear so to the student at this point.

2.2 Solving Linear Systems by Row Reduction Considerations in Solving Linear Systems • Echelon Forms • General Solutions as Linear Combinations of Column Vectors • Gauss-Jordan and Gaussian Elimination • Some Facts About Echelon Forms • Back Substitution • Homogeneous Linear Systems • The Dimension Theorem for Homogeneous Linear Systems • Stability, Roundoff Error, and Partial Pivoting

Comments on 2.2: The basic methods for solving linear systems are discussed. Example 4 uses the material in Section 1.3 to analyze the geometric structure of solution sets. Theorems 2.2.2 and 2.2.3 are important should be emphasized. If you want to focus on numerical issues, then you may want to devote some class time to the subsection on roundoff error on p.58 and you may want to assign Exercises 53 and/or 54.

2.3 Applications of Linear Systems Global Positioning • Network Analysis • Electrical Circuits • Balancing Chemical Equations • Polynomial Interpolation

Comments on 2.3: This is not a core section, so what you cover from will depend on your available time. The applications are independent of one another, so you are free to pick and choose.

CHAPTER 3 MATRICES AND MATRIX ALGEBRA

3.1 Operations on Matrices Matrix Notation and Terminology • Operations on Matrices • Row and Column Vectors • The Product Ax • The Product AB • Finding Specific Entries in a Matrix Product • Finding Specific Rows and Columns of a Matrix Product • Matrix Products as Linear Combinations • Transpose of a Matrix • Trace • Inner and Outer Matrix Products

Comments on 3.1: In keeping with the recommendations of the Linear Algebra Curriculum Study Group, matrix multiplication is viewed in various ways: Definition 3.1.4 defines the product Ax as a linear combination of the columns of A, Theorem 3.1.7 considers the traditional entry by entry method of computing AB, Formulas (18) and (19) focus on computing specific rows or columns of a matrix product, and Theorem 3.1.8 views matrix products as linear combinations. Formulas (25), (26), and (27), which related dot products and matrix multiplication, will prove to be important and should not be omitted.

3.2 Inverses; Algebraic Properties of Matrices Properties of Matrix Addition and Scalar Multiplication • Properties of Matrix Multiplication • Zero Matrices • Identity Matrices • Inverse of a Matrix • Properties of Inverses • Powers of a Matrix • Matrix Polynomials • Properties of the Transpose • Properties of the Trace • Transpose and Dot Product

Comments on 3.2: Although all of the material in this section is important, there is a lot of detail, so you will have to use your class time judiciously if you want to cover the section in a single lecture. You might consider focusing on the material through the end of Example 9 (on p. 101) in class (but omit Example 8) and then just summarize the rest of the material. Alternatively, you could devote an additional half lecture to this material, and make up the time by devoting less than a full lecture to Sections 3.6 and/or 3.8, both of which are less demanding.

3.3 Elementary Matrices; A Method for Finding the Inverse of a Matrix Elementary Matrices • Characterizations of Invertibility • Row Equivalence • An Algorithm for Inverting Matrices • Solving Linear Systems by Matrix Inversion • Solving Multiple Linear Systems with a Common Coefficient Matrix • Consistency of Linear Systems

Comments on 3.3: All of the material in this section is important. Theorem 3.3.9 on deserves special emphasis since it unifies the most important ideas in the section. This theorem will become a theme as the text progresses, and new parts will be added as new ideas are developed.

3.4 Subspaces and Linear Independence Subspaces of Rn • Solution Space of a Linear System • Linear Independence • Linear Independence and Homogeneous Linear Systems • Translated Subspaces • A Unifying Theorem

Comments on 3.4: All of the material in this section is important. You may want to spend a few minutes of class time on the "Looking Ahead" discussion on p.125. Example 5 is important in that it illustrates that the solution space of a linear system is viewed as the span of a set of vectors. One of the main pedagogical goals of this text is to teach the student how to look at concepts in various ways and this is a first step. The unifying Theorem 3.4.9 provides a useful lecture summary.

3.5 The Geometry of Linear Systems The Relationship Between Ax = b and Ax = 0 • Consistency of a Linear System from the Vector Point of View • Hyperplanes • Geometric Interpretations of Solution Spaces

Comments on 3.5: All of the material in this section is important. This exposition reflects our strong emphasis on geometric visualization. The discussion on p.138 in which consistency of a linear system is viewed from a vector point of view deserves emphasis, since it teaches the student how to change points of view. Theorem 3.5.6, which is not obvious, should be emphasized since it will become important later in the text. On p.140 we take our first informal look at the concept of dimension. The concluding table on p. 140 looks at the notion of dimension in the context of homogeneous linear systems both algebraically and geometrically.

3.6 Matrices with Special Forms Diagonal Matrices • Triangular Matrices • Linear Systems with Triangular Coefficient Matrices • Properties of Triangular Matrices • Symmetric and Skew-Symmetric Matrices • Invertibility of Symmetric Matrices • Products of A and its Transpose • Fixed Points of a Matrix • A Technique for Inverting A When A Is Nilpotent • Inverting I-A by Power Series

Comments on 3.6: If desired, you can omit the material that follows Example 6. It is only used in the discussion of Theorem 5.2.1 on Leontief models and can be discussed within the context of that theorem if you cover Section 5.2. If you need to pick up some time, you can cover this section in less than one lecture.

3.7 Matrix Factorizations; LU-Decomposition Solving Linear Systems by Factorization • Finding LU-Decompositions • The Relationship Between Gaussian Elimination and LU-Decomposition • Matrix Inversion by LU-Decomposition • LU-Decompositions • Using Permutation Matrices to Deal with Row Interchanges • Flops and the Cost of Solving a Linear System • Cost Estimates for Solving Large Linear Systems • Considerations in Choosing an Algorithm for Solving a Linear System

Comments on 3.7: If desired, you can omit the material that starts with the subsection "Flops and the Cost of Solving A Linear System" on p.160 and ends with Table 3.7.1 on p.163. If you do so, then you may want to comment briefly on Table 3.7.1.

3.8 Partitioned Matrices and Parallel Processing General Partitioning • Block Diagonal Matrices • Block Upper Triangular Matrices

Comments on 3.8: Partitioned matrices now play such an important role in applications that we have made this section part of the core material even though we are aware that this material is often omitted by many instructors. If you need to pick up some time to hold your schedule, then you can cover this section in less than one lecture.

CHAPTER 4 DETERMINANTS

4.1 Determinants; Cofactor Expansion Determinants of 2 x 2 and 3 x 3 Matrices • Elementary Products • General Determinants • Evaluation Difficulties for Higher-Order Determinants • Determinants of Matrices with Rows or Columns That Have All Zeros • Determinants of Triangular Matrices • Minors and Cofactors • Cofactor Expansions

Comments on 4.1: All of the material in this section is important. Depending on your teaching style and the background of your students, you may not need an entire lecture for this section.

4.2 Properties of Determinants Determinant of the Transpose of a Matrix • Effect of Elementary Row Operations on a Determinant • Simplifying Cofactor Expansions • Determinants by Gaussian Elimination • A Determinant Test for Invertibility • Determinant of a Product of Matrices • Determinant Evaluation by LU-Decomposition • Determinant of the Inverse of a Matrix • Determinant of A + B • A Unifying Theorem

Comments on 4.2: If you have omitted Section 3.7 on LU-decompositions, then omit the subsection "Determinant Evaluation by LU-Dectomposition on p.189.

4.3 Cramer's Rule; Formula for the Inverse of a Matrix; Applications of Determinants Adjoint of a Matrix • A Formula for the Inverse of a Matrix • How the Inverse Formula Is Actually Used • Cramer's Rule • Geometric Interpretation of Determinants • Polynomial Interpolation and the Vandermonde Determinant • Cross Products

Comments on 4.3: If you want to cover determinants in two lectures, rather than the three we have allocated in the 35-lecture core, then cover 4.1 in half a lecture and cover this section in half a lecture by stopping at the end of Example 6. However, this will eliminate the material on cross products, a little of which is used in the study of rotations. The facts needed for the study of rotations are that u x v is orthogonal to u and v and is determined by the right-hand rule.

4.4 A First Look at Eigenvalues and Eigenvectors Fixed Points • Eigenvalues and Eigenvectors • Eigenvalues of Triangular Matrices • Eigenvalues of Powers of a Matrix • A Unifying Theorem • Complex Eigenvalues • Algebraic Multiplicity • Eigenvalue Analysis of 2 by 2 Matrices • Eigenvalue Analysis of 2 by 2 Symmetric Matrices • Expressions for Determinant and Trace in Terms of Eigenvalues • Eigenvalues by Numerical Methods

Comments on 4.4: This section covers most of the basic ideas about eigenvectors and eigenvalues. Eigenvalues and eigenvectors appear again in Section 5.4 (the power method and its application to internet search engines) and in the study of diagonalizability in Chapter 8. Theorem 4.4.7 provides a useful lecture summary.

CHAPTER 5 MATRIX MODELS

5.1 Dynamical Systems and Markov Chains Dynamical Systems • Markov Chains • Markov Chains as Powers of the Transition Matrix • Long-Term Behavior of a Markov Chain

Comments on 5.1: Although the idea of a "limit" occurs in this section, we have presented it in a way that calculus is not required. We also give a brief informal discussion of "probability" to make the section self contained.

5.2 Leontief Input-Output Models Inputs and Outputs in an Economy • The Leontief Model of an Open Economy • Productive Open Economies

Comments on 5.2: You can shorten the section and still give the student the "flavor" of the application by omitting the subsection on "Productive Open Economies" (starting on p.238).

5.3 Gauss--Seidel and Jacobi Iteration; Sparse Linear Systems Iterative Methods • Jacobi Iteration • Gauss--Seidel Iteration • Convergence • Speeding Up Convergence

Comments on 5.3: The procedures followed for handling roundoff can affect the numerical values obtained in examples and exercises. You might want to advise your students to use the full accuracy of the calculating device and to avoid rounding at intermediate steps.

5.4 The Power Method; Application to Internet Search Engines The Power Method • The Power Method with Euclidean Scaling • The Power Method with Maximum Entry Scaling • Rate of Convergence • Stopping Procedures • An Application of the Power Method to Internet Searches • Variations of the Power Method

Comments on 5.4: The procedures followed for handling roundoff can affect the numerical values obtained in examples and exercises. You might want to advise your students to use the full accuracy of the calculating device and to avoid rounding at intermediate steps.

CHAPTER 6 LINEAR TRANSFORMATIONS

6.1 Matrices as Transformations A Review of Functions • Matrix Transformations • Linear Transformations • Some Properties of Linear Transformations • All Linear Transformations from Rn to Rm Are Matrix Transformations • Rotations About the Origin • Reflections About Lines Through the Origin • Orthogonal Projections onto Lines Through the Origin • Transformations of the Unit Square • Power Sequences

Comments on 6.1: All of the material in this section is important. We have introducted the concept of a linear transformation using a simple Hooke's law illustration from physics.

6.2 Geometry of Linear Operators Norm-Preserving Linear Operators • Orthogonal Operators Preserve Angles and Orthogonality • Orthogonal Matrices • All Orthogonal Linear Operators on R2 are Rotations or Reflections • Contractions and Dilations of R2 • Vertical and Horizontal Compressions and Expansions of R2 • Shears • Linear Operators on R3 • Reflections About Coordinate Planes • Rotations in R3 • General Rotations

Comments on 6.2: There is a fair amount of material in this section, so you will need to use your class time judiciously if you want to cover the material in one lecture. We suggest that you not spend a lot of time on Tables 6.2.1, 6.2.2, 6.2.3, and 6.2.4. Once the student understands the idea, the idea is repetitive. The tables are provided for reference. If desired, you can codense the presentation of general rotations in R3 by focusing primarily on rotations about the coordinate axes and Table 6.2.6.

6.3 Kernel and Range Kernel of a Linear Transformation • Kernel of a Matrix Transformation • Range of a Linear Transformation • Range of a Matrix Transformation • Existence and Uniqueness Issues • One-to-One and Onto from the Viewpoint of Linear Systems • A Unifying Theorem

Comments on 6.3: All of the material in this section is important. The unifying theorem 6.3.15 provides a useful lecture summary.

6.4 Composition and Invertibility of Linear Transformations Compositions of Linear Transformations • Compositions of Three or More Linear Transformations • Factoring Linear Operators into Compositions • Inverse of a Linear Transformation • Invertible Linear Operators • Geometric Properties of Invertible Linear Operators on R2 • Image of the Unit Square Under an Invertible Linear Operator

Comments on 6.4: You can shorten this section by eliminating one or more of the subsections: "Factoring Linear Operators into Compositions" (starting on p.310), "Geometric Properties of Linear Operators on R2 (starting on p.314), and "Image of the Unit Square Under an Invertible Linear Operator" (p.315).

6.5 Computer Graphics Wireframes • Matrix Representations of Wireframes • Transforming Wireframes • Translation Using Homogeneous Coordinates • Three-Dimensional Graphics

Comments on 6.5: You can shorten this section by eliminating the subsection on "Three-Dimensional Graphics" (starting on p.323).

CHAPTER 7 DIMENSION AND STRUCTURE

7.1 Basis and Dimension Bases for Subspaces • Dimension of a Solution Space • Dimension of a Hyperplane

Comments on 7.1: We introduce the idea of a hyperplane (p.333), which is not discussed in most basic texts. This generalization of lines in R2 and planes in R3 will be useful in studying the geometry of linear systems and geometric properties of Rn. They will also help your students develop their geometric intuition about n-space.

7.2 Properties of Bases Properties of Bases • Subspaces of Subspaces • Sometimes Spanning Implies Linear Independence and Conversely • A Unifying Theorem

Comments on 7.2: The unifying theorem 7.2.7 provides a useful lecture summary.

7.3 The Fundamental Spaces of a Matrix The Fundamental Spaces of a Matrix • Orthogonal Complements • Properties of Orthogonal Complements • Finding Bases by Row Reduction • Determining Whether a Vector Is in a Given Subspace

Comments on 7.3: You might draw the student's attention to the relationship between Example 6 in this section to the Consistency Problem 3.3.10 posed in Chapter 3 on p.118, and Example 8 on p.119.

7.4 The Dimension Theorem and Its Implications The Dimension Theorem for Matrices • Extending a Linearly Independent Set to a Basis • Some Consequences of the Dimension Theorem for Matrices • The Dimension Theorem for Subspaces • A Unifying Theorem • More on Hyperplanes • Rank 1 Matrices • Symmetric Rank 1 Matrices

Comments on 7.4: If you are really pressed for time you can shorten this section by eliminating the material on rank 1 matrices (starting on p. 355), though we advise against it. There are some brief references to this material in Sections 7.7 and 8.3 (see Theorem 7.7.3 and the Remark following Example 2 on p.472)., but you can refer the student to the deleted rank 1 material at the appropriate time. The unifying theorem 7.4.4 provides a useful lecture summary.

7.5 The Rank Theorem and Its Implications The Rank Theorem • Relationship Between Consistency and Rank • Overdetermined and Underdetermined Linear Systems • Products of A and it Transpose • Some Unifying Theorems • Applications of Rank

Comments on 7.5: Emphasize that Theorem 7.5.10 applies to all matrices, whereas the previous unifying theorems such as 7.4.4 apply only to square matrices.

7.6 The Pivot Theorem and Its Implications Basis Problems Revisited • Bases for the Fundamental Spaces of a Matrix • A Column-Row Factorization • Column-Row Expansion

Comments on 7.6: You can shorten this section by eliminating the subsections "A Column-Row Factorization" and "Column-Row Expansion" (starting on p.375).

7.7 The Projection Theorem and Its Implications Orthogonal Projections onto Lines Through the Origin in R2 • Orthogonal Projections onto Lines Through the Origin in Rn • Projection Operators on Rn • Orthogonal Projections onto General Subspaces • When Does a Matrix Represent an Orthogonal Projection? • Strang Diagrams • Full Column Rank and Consistency of a Linear System • The Double Perp Theorem • Orthogonal Projections onto an Orthogonal Complement

Comments on 7.7: If your goal is to reach the material on fitting curves to experimental data (p.399) without "frills", then this section and the next can be shortened considerably: One possibility is to eliminate all of the material after Example 6 in this section and skip the proofs of parts (b) and (c) of Theorem 7.8.3 in the next section.

7.8 Best Approximation and Least Squares Minimum Distance Problems • Least Squares Solutions of Linear Systems • Finding Least Squares Solutions of Linear Systems • Orthogonality Property of Least Squares Error Vectors • Strang Diagrams for Least Squares Problems • Fitting a Curve to Experimental Data • Least Squares Fits by Higher-Degree Polynomials • Theory Versus Practice

Comments on 7.8: See the comments on Section 7.7.

7.9 Orthonormal Bases and the Gram-Schmidt Process Orthogonal and Orthonormal Bases • Orthogonal Projections Using Orthogonal Bases • Trace and Orthogonal Projections • Linear Combinations of Orthonormal Basis Vectors • Finding Orthogonal and Orthonormal Bases • A Property of the Gram--Schmidt Process • Extending Orthonormal Sets to Orthonormal Bases

Comments on 7.9: In this day and age there is little to be gained by having the student do laborious computations such as those in Example 6. Once the student understands the underlying idea and the Gram-Schmidt algorithm, it would make sense to take advantage of technology, if available.

7.10 QR-Decomposition; Householder Transformations QR-Decomposition • The Role of QR-Decomposition in Least Squares Problems • Other Numerical Issues • Householder Reflections • QR-Decomposition Using Householder Reflections • Householder Reflections in Applications

Comments on 7.10: If you are not focusing extensively on numerical matters in your course, then you can shorten this section by eliminating all of the material on Householder reflections and transformations (starting on p.420).

7.11 Coordinates with Respect to a Basis Nonrectangular Coordinate Systems in R2 and R3 • Coordinates with Respect to an Orthonormal Basis • Computing with Coordinates with Respect to an Orthononormal Basis • Change of Basis for Rn • Invertibility of Transition Matrices • A Good Technique for finding Transition Matrices • Coordinate Maps • Transition Between Orthonormal Bases • Application to Rotation of Coordinate Axes • New Ways to Think About Matrices

Comments on 7.11: Change-of-basis computations are always bothersome to beginning students. We think that Formula (24) and the highlighted procedure that follows it will be a big help to your students.

CHAPTER 8 DIAGONALIZATION

8.1 Matrix Representations of Linear Transformations - Matrix of a Linear Operator with Respect to a Basis • Changing Bases • Matrix of a Linear Transformation with Respect to a Pair of Bases • Effect of Changing Bases on Matrices of Linear Transformations • Representing Linear Operators with Two Bases

Comments on 8.1: Students traditionally have trouble with the computations in this section. These difficulties can be minimized by having the students learn to construct the kinds of "communtative diagrams" in Figures 8.1.3 and 8.1.7. This will really prove helpful when different notations are used for bases and transformations.

8.2 Similarity and Diagonalizability Similar Matrices • Similarity Invariants • Eigenvectors and Eigenvalues of Similar Matrices • Diagonalization • A Method for Diagonalizing a Matrix • Linear Independence of Eigenvectors • Relationship Between Algebraic and Geometric Multiplicity • A Unifying Theorem on Diagonalizability

Comments on 8.2: The remarks on p.459 and p.462 are worth emphasizing.

8.3 Orthogonal Diagonalizability; Functions of a Matrix Orthogonal Similarity • A Method for Orthogonally Diagonalizing a Symmetric Matrix • Spectral Decomposition • Powers of a Diagonalizable Matrix • Cayley-Hamilton Theorem • Exponential of a Matrix • Diagonalization and Linear Systems • The Nondiagonalizable Case

Comments on 8.3: You can shorten this section by stopping at the end of Example 5, thereby eliminating Shur's Theorem and Hessenberg's Theorem. However, if you plan to cover Section 8.6 on Singular Value Decomposition, then you should touch on these theorems briefly, since they are referred to in that the introductory part of that section. You can shorten this section further by eliminating the subsection "Exponential of a Matrix". If you do this, then in Section 8.10 on Systems of Differential Equations you will want to stop at the end of Example 6. Finally, the subsections on "Powers of a Matrix" and the "Cayley-Hamilton Theorem" are actually independent of the other material in the section, so if time is a serious issue you can teach one or both of those subsections as a continuation of Section 8.2 and eliminate all of the other material in this section.

8.4 Quadratic Forms Definition of a Quadratic Form • Change of Variable in a Quadratic Form • Quadratic Forms in Geometry • Identifying Conic Sections • Positive Definite Quadratic Forms • Classifying Conic Sections Using Eigenvalues • Identifying Positive Definite Matrices • Cholesky Factorization

Comments on 8.4: You can shorten this section by stopping with the Concept Problem following Example 5 on p.491.

8.5 Application of Quadratic Forms to Optimization Relative Extrema of Functions of Two Variables • Constrained Extremum Problems • Constrained Extrema and Level Curves

Comments on 8.5: This section requires calculus. You may want to compare the method used in Example 2 and 3 to other methods studied in calculus (Lagrange multipliers, for example).

8.6 Singular Value Decomposition Singular Value Decomposition of Square Matrices • Singular Value Decomposition of Symmetric Matrices • Polar Decomposition • Singular Value Decomposition of Nonsquare Matrices • Singular Value Decomposition and the Fundamental Spaces of a Matrix • Reduced Singular Value Decomposition • Data Compression and Image Processing • Singular Value Decomposition from the Transformation Point of View

Comments on 8.6: Although singular value decompositions are extremely important in contemporary applications of linear algebra, this material will be difficult for many students. You can shorten this section by eliminating the final subsection on "Singular Value Decomposition from the Transformation Point of View".

8.7 The Pseudoinverse The Pseudoinverse • Properties of the Pseudoinverse • The Pseudoinverse and Least Squares • Condition Number and Numerical Considerations

Comments on 8.7: The notion of a pseudoinverse is so ubiquitous in the contemporary literature that you may want to devote five or ten minutes of class time to explain what the section is about, even if you are not covering it.

8.8 Complex Eigenvalues and Eigenvectors Vectors in Complex n-Space • Algebraic Properties of the Complex Conjugate • The Complex Euclidean Inner Product • Vector Space Concepts in Cn • Complex Eigenvalues of Real Matrices Acting on Vectors in Complex n-space • A Proof That Real Symmetric Matrices Have Real Eigenvalues • A Geometric Interpretation of Complex Eigenvalues of Real Matrices

Comments on 8.8: Although we have not included this section as part of the core, we think it should be high on your list of sections to be added to the core if time is available. For one thing, one cannot understand why real symmetric matrices have real eigenvalues with some knowledge of Cn (see Theorem 8.8.7). Also, the material in this section clarifies the geometric significance of complex eigenvalues of real matrices in the 2x2 case (see Theorem 8.8.8 and the related material that follows it).

8.9 Hermitian, Unitary, and Normal Matrices Hermitian and Unitary Matrices • Unitary Diagonalizability • Skew-Hermitian Matrices • Normal Matrices • A Comparison of Eigenvalues

Comments on 8.9: If you do not plan to cover this section, you might still consider devoting 15 minutes of class time to the terminology and to the discussion relating to Figure 8.9.1.

8.10 Systems of Differential Equations Terminology • Linear Systems of Differential Equations • Fundamental Solutions • Solutions Using Eigenvalues and Eigenvectors • Exponential Form of a Solution • The Case Where A is Not Diagonalizable

Comments on 8.10: You can shorten this section by stopping at the end of Example 6.

CHAPTER 9 GENERAL VECTOR SPACES

9.1 Vector Space Axioms Vector Space Axioms • Function Spaces • Matrix Spaces • Unusual Vector Spaces • Subspaces • Linear Independence, Spanning, Basis • Wronski's Test for Linear Independence of Functions • Dimension • The Lagrange Interpolating Polynomials • Lagrange Interpolation from a Vector Point of View

Comments on 9.1: We recommend keeping this section in its current position for pedagogical reasons, but if you disagree and would like to teach it earlier, then place it after Section 7.2. Also, you can shorten this section by eliminating the material on Lagrange interpolating polynomials (starting on p.564). However, if you have time to include that material then you will have a useful and nontrivial example of a basis for the space of polynomials of degree n-1 or less (Theorem 9.1.9).

9.2 Inner Product Spaces; Fourier Series Inner Product Axioms • The Effect of Weighting on Geometry • Algebraic Properties of Inner Products • Orthonormal Bases • Best Approximation • General Inner Products on Rn

Comments on 9.2: We recommend keeping this section in its current position for pedagogical reasons, but if you disagree and would like to teach it earlier, then teach it right after Section 9.1 but eliminate the subsection "General Inner Products" if you have not covered postive definite matrices (Section 8.4).

Errata: In Technology Exercise T2, The weighted Euclidean Inner Product should be <u, v> = 3u1v1 + 2u2v2 + u3v3.

9.3 General Linear Transformations; Isomorphism

Comments on 9.3: We recommend keeping this section in its current position for pedagogical reasons, but if you disagree and would like to teach it earlier, then teach it right after Section 9.2.

APPENDIX A HOW TO READ THEOREMS Contrapositive Form of a Theorem • Converse of a Theorem • Theorems Involving Three or More Implications

APPENDIX B COMPLEX NUMBERS Complex Numbers • The Complex Plane • Polar Form of a Complex Number • Geometric Interpretation of Multiplication and Division of Complex Numbers • DeMoivre's Formula • Euler's Formula