Mathematical Methods I
Mathematical Methods I
University of Notre Dame, Fall 2019
M W F 10:30 am  11:20 am (Lectures, DeBartolo 129)
Professor Nicholas Zabaras
Lecture notes and videos
 Vector and Matrices
 Linear combination of vectors, inner product, length, angle between vectors; Different views of Ax=b.
[VideoLecture] [Lecture Notes]
 Linear combination of vectors, inner product, length, angle between vectors; Different views of Ax=b.

Solving Ax=b, Linear Independence, Rank, Column, Row and Null Spaces. When can I solve Ax=b? Basis and Linear Independence; Matrix operations, symmetric, skew, triangular and diagonal matrices; Rank of a matrix, rank and linear independence; Vector spaces and subspaces, basis, span; A first look at the column space, nullspace and row space; Inner product spaces, CauchyScwartz inequality.
[VideoLecture] [Lecture Notes]
 When can I solve Ax=b? Basis and Linear Independence; Matrix operations, symmetric, skew, triangular and diagonal matrices; Rank of a matrix, rank and linear independence; Vector spaces and subspaces, basis, span; A first look at the column space, nullspace and row space; Inner product spaces, CauchyScwartz inequality.

Rank, Column, Row and Null Spaces Rank of a matrix, rank and linear independence; Vector spaces and subspaces, basis, span; A first look at the column space, nullspace and row space; Inner product spaces, CauchyScwartz inequality.
[VideoLecture] [Lecture Notes]
 Rank of a matrix, rank and linear independence; Vector spaces and subspaces, basis, span; A first look at the column space, nullspace and row space; Inner product spaces, CauchyScwartz inequality.

Matrix Operations and Elementary Matrices Review of the column and row views of Ax=b; Matrix Multiplication, Elementary Matrices, Permutation Matrices, Exchanging Columns, Multiplication as a Sum of Rank 1 Matrices, Matrix Block Multiplication; Matrix Inversion, Cancellation Laws of Matrix Multiplication, Inverse and Solution of Ax=b; Summary of Matrix Operations
[VideoLecture] [Lecture Notes]
 Review of the column and row views of Ax=b; Matrix Multiplication, Elementary Matrices, Permutation Matrices, Exchanging Columns, Multiplication as a Sum of Rank 1 Matrices, Matrix Block Multiplication; Matrix Inversion, Cancellation Laws of Matrix Multiplication, Inverse and Solution of Ax=b; Summary of Matrix Operations

Matrix Elimination Matrix Elimination, the augmented matrix, elimination example, elimination matrices and their inverses, elimination with row exchange; A=LU factorization.
[VideoLecture] [Lecture Notes]
 Matrix Elimination, the augmented matrix, elimination example, elimination matrices and their inverses, elimination with row exchange; A=LU factorization.

LU Factorization, Matrix Inversion, GaussJordan Form A=LU factorization, A=LDU factorization, forward elimination and back substitution, Cost of factorization, Permutation matrices and row exchanges, row equivalent systems, Schur Complement; Row Echelon form and when does elimination fails? Inverse of a matrix, Gauss – Jordan form.
[VideoLecture] [Lecture Notes]
 A=LU factorization, A=LDU factorization, forward elimination and back substitution, Cost of factorization, Permutation matrices and row exchanges, row equivalent systems, Schur Complement; Row Echelon form and when does elimination fails? Inverse of a matrix, Gauss – Jordan form.

Vector Spaces and Subspaces, Basis and Dimension Spaces of vectors, the space ℝ^{n}, vector space, examples of vector spaces, the zero vector; Subspace, union of subspaces, subspaces of ℝ^{3}, examples; The column space of A, Subspace spanned by a set of vectors, examples; The null space of A, solving Ax=0 and Rx=0, examples, pivot columns and free columns, the reduced row echelon form, solving Ax=0 and special solutions, computing the nullspace; The rank of a matrix, rank 1 matrices, rank as the dimension of C(A); Matrix elimination summary.
[VideoLecture] [Lecture Notes]
 Spaces of vectors, the space ℝ^{n}, vector space, examples of vector spaces, the zero vector; Subspace, union of subspaces, subspaces of ℝ^{3}, examples; The column space of A, Subspace spanned by a set of vectors, examples; The null space of A, solving Ax=0 and Rx=0, examples, pivot columns and free columns, the reduced row echelon form, solving Ax=0 and special solutions, computing the nullspace; The rank of a matrix, rank 1 matrices, rank as the dimension of C(A); Matrix elimination summary.

Lecture 8; The column space of A; The null space of A; The row space of A; The nullspace of A^{T}; Solving Ax=0; Rank of a matrix; The complete solution to Ax=b; Independence, basis and dimension; Dimensions of the four fundamental spaces
[VideoLecture] [Lecture Notes]
 The column space of A; The null space of A; The row space of A; The nullspace of A^{T}; Solving Ax=0; Rank of a matrix; The complete solution to Ax=b; Independence, basis and dimension; Dimensions of the four fundamental spaces

Independence, Basis, and Dimension Independence, basis and dimension, vectors that span a subspace, row and column spaces, bases for the column and row spaces, dimension of a vector space, dimensions of C(A) and N(A), bases for matrix and function spaces; Summary – independence, basis and dimension.
[VideoLecture] [Lecture Notes]
 Independence, basis and dimension, vectors that span a subspace, row and column spaces, bases for the column and row spaces, dimension of a vector space, dimensions of C(A) and N(A), bases for matrix and function spaces; Summary – independence, basis and dimension.

Dimensions of the Four Fundamental Subspaces The four subspaces of A, C(A), N(A), C(A^{T}), N(A^{T}); The dimensions of the column, row, nullspace and left null spaces; Applications.
[VideoLecture] [Lecture Notes]
 The four subspaces of A, C(A), N(A), C(A^{T}), N(A^{T}); The dimensions of the column, row, nullspace and left null spaces; Applications.

The Four Fundamental Subspaces: Applications to Graphs and Orthogonality
 An example of the fundamental subspaces from graphs, networks and electrical circuits; Example in rank 1 matrices and in rank 2 matrices; Summary of dimensions and bases of the four fundamental subspaces; Inner Product Spaces, Orthogonality, Orthogonality of the four fundamental subspaces, Orthogonal complements, Combining spaces.
[VideoLecture] [Lecture Notes]
 An example of the fundamental subspaces from graphs, networks and electrical circuits; Example in rank 1 matrices and in rank 2 matrices; Summary of dimensions and bases of the four fundamental subspaces; Inner Product Spaces, Orthogonality, Orthogonality of the four fundamental subspaces, Orthogonal complements, Combining spaces.
 Projections and Least Squares
 Projections matrices, Projection onto a line, Projection onto a subspace, Normal equations of least squares.
[VideoLecture] [Lecture Notes]
 Projections matrices, Projection onto a line, Projection onto a subspace, Normal equations of least squares.

Projections and Least Squares (Continued) Projections matrices, Projection onto a line, Projection onto a subspace, Summary; Least Squares approximation, Minimizing the error, Geometric presentation of least squares, Regression – Fitting a straight line, Fitting a parabola, Summary; Orthogonal basis and GramSchmidt orthogonalization, Projections using an orthonormal basis, The GramSchmidt Decomposition, The 𝑨=𝑸𝑹 factorization, Summary.
[VideoLecture] [Lecture Notes]
 Projections matrices, Projection onto a line, Projection onto a subspace, Summary; Least Squares approximation, Minimizing the error, Geometric presentation of least squares, Regression – Fitting a straight line, Fitting a parabola, Summary; Orthogonal basis and GramSchmidt orthogonalization, Projections using an orthonormal basis, The GramSchmidt Decomposition, The 𝑨=𝑸𝑹 factorization, Summary.

Least Squares, Orthogonal Matrices, GramSchmidt and A=QR Factorization (Continued) Reviewing projection onto a subspace, Least Squares approximation, Minimizing the error, Linear regression; Orthogonal basis and GramSchmidt orthogonalization, Projections using an orthonormal basis, The A=QR factorization, Summary.
[VideoLecture] [Lecture Notes]
 Reviewing projection onto a subspace, Least Squares approximation, Minimizing the error, Linear regression; Orthogonal basis and GramSchmidt orthogonalization, Projections using an orthonormal basis, The A=QR factorization, Summary.

The Properties of Determinants
 detI=1, sign reversal and linearity in each row; Elimination and determinant; Invertable matrices and determinant; The determinant of AB and of A^{T}; Permutations and cofactors; The Pivot Formula.
[VideoLecture] [Lecture Notes]
 detI=1, sign reversal and linearity in each row; Elimination and determinant; Invertable matrices and determinant; The determinant of AB and of A^{T}; Permutations and cofactors; The Pivot Formula.

Determinant Formulas, Cofactors, Cramer's Rule, Inverse Matrix, Area and Volume Formulas for evaluating the determinant; Cofactor formula; Evaluating A^{1}; Cramer's rule for x=A^{1}b; Area and Volume as Determinants.
[VideoLecture] [Lecture Notes]
 Formulas for evaluating the determinant; Cofactor formula; Evaluating A^{1}; Cramer's rule for x=A^{1}b; Area and Volume as Determinants.

Introduction to Eigenvalues and Eigenvectors Introduction to Eigenvalues and Eigenvectors; Determining eigenvalues and eigenvectors; Applications to Symmetric and Orthogonal Matrices; The Eigenvalue Problem for Projection and Reflection Matrices; Introduction to Diagonalization of a Matrix.
[VideoLecture] [Lecture Notes]
 Introduction to Eigenvalues and Eigenvectors; Determining eigenvalues and eigenvectors; Applications to Symmetric and Orthogonal Matrices; The Eigenvalue Problem for Projection and Reflection Matrices; Introduction to Diagonalization of a Matrix.
 Diagonalization of a Matrix, Matrix Exponential
 Distinct Eigenvalues, Independent Eigenvectors and Diagonalization of a General Matrix; Examples of Diagonalizable and NonDiagonalizable Matrices; Markov Matrices; Similar Matrices; Example of Fibonacci Numbers; Geometric and Algebraic Multiplicity.
[VideoLecture] [Lecture Notes]
 Distinct Eigenvalues, Independent Eigenvectors and Diagonalization of a General Matrix; Examples of Diagonalizable and NonDiagonalizable Matrices; Markov Matrices; Similar Matrices; Example of Fibonacci Numbers; Geometric and Algebraic Multiplicity.

Differential Equations and the Eigenvalue Problem, Exponential of a Matrix, Diagonalization of Symmetric Matrices System of Differential Equations, Difference Equations, Stability; Exponential of a Matrix, Examples; Symmetric and AntiSymmetric Matrices, Diagonalization and Spectral Theorem; Repeated eigenvalues and Diagonalization; Schnur Theorem.
[VideoLecture] [Lecture Notes]
 System of Differential Equations, Difference Equations, Stability; Exponential of a Matrix, Examples; Symmetric and AntiSymmetric Matrices, Diagonalization and Spectral Theorem; Repeated eigenvalues and Diagonalization; Schnur Theorem.

Positive Definite Matrices and the Jordan Form Positive Definite Matrices; Tests for positive definite matrices  eigenvalues, pivots, subdeterminants, energy functional, A^{T}A form; Positive SemiDefinite Matrices; Test for a Minimum of Energy; Quadratic form, transformation to principal axes; Similar matrices and Jordan form; Similarity Transformation.
[VideoLecture] [Lecture Notes]
 Positive Definite Matrices; Tests for positive definite matrices  eigenvalues, pivots, subdeterminants, energy functional, A^{T}A form; Positive SemiDefinite Matrices; Test for a Minimum of Energy; Quadratic form, transformation to principal axes; Similar matrices and Jordan form; Similarity Transformation.

Singular Value Decomposition Low rank image representation; Eigenvectors for the SVD; Singular Values; Proof of the SVD; Examples of the SVD; Accounting for the nullspaces; Singular Value Stability versus Eigenvalue Stability.
[VideoLecture] [Lecture Notes]
 Low rank image representation; Eigenvectors for the SVD; Singular Values; Proof of the SVD; Examples of the SVD; Accounting for the nullspaces; Singular Value Stability versus Eigenvalue Stability.
 Linear Transformations
 Linear Transformations, Matrix Representation; Basis; Examples of linear transformations; Range and Kernel; Product of transformations; Choosing the best bases; Jordan form; Bases for Function spaces.
[VideoLecture] [Lecture Notes]
 Linear Transformations, Matrix Representation; Basis; Examples of linear transformations; Range and Kernel; Product of transformations; Choosing the best bases; Jordan form; Bases for Function spaces.
 Introduction to 1st Order Ordinary Differential Equations
 Differential Equation Models; Ordinary and Partial Differential Equations; Classification of ODEs; Direction fields and Equilibrium solutions; Solutions of simple ODEs; Initial Value Problems; Method of Integrating factors; Integrating factors for general 1st Order ODEs.
[VideoLecture] [Lecture Notes]
 Differential Equation Models; Ordinary and Partial Differential Equations; Classification of ODEs; Direction fields and Equilibrium solutions; Solutions of simple ODEs; Initial Value Problems; Method of Integrating factors; Integrating factors for general 1st Order ODEs.

1st Order Ordinary Differential Equations Direction fields and Equilibrium solutions; Solutions of ODEs; Separable ODEs, Method of Integrating Factors; Exact ODEs.
[VideoLecture] [Lecture Notes]
 Direction fields and Equilibrium solutions; Solutions of ODEs; Separable ODEs, Method of Integrating Factors; Exact ODEs.

Linear Vs. Nonlinear ODEs, Autonomous Eqs and Population Dynamics, Existence and Uniqueness of Solutions Difference between linear and nonlinear Equations; Autonomous Equations and Population Dynamics; Numerical Approximations, Euler Method; Existence and Uniqueness of Solutions for linear and nonlinear 1st Order ODEs.
[VideoLecture] [Lecture Notes]
 Difference between linear and nonlinear Equations; Autonomous Equations and Population Dynamics; Numerical Approximations, Euler Method; Existence and Uniqueness of Solutions for linear and nonlinear 1st Order ODEs.

2nd Order Differential Equations SecondOrder Linear Homogeneous Equations with Constant Coefficients, Solutions of Linear Homogeneous Equations; The Wronskian; Complex Roots of the Characteristic Equation, Repeated Roots; Reduction of Order, Nonhomogeneous Equations; Method of Undetermined Coefficients, Variation of Parameters.
[VideoLecture] [Lecture Notes]
 SecondOrder Linear Homogeneous Equations with Constant Coefficients, Solutions of Linear Homogeneous Equations; The Wronskian; Complex Roots of the Characteristic Equation, Repeated Roots; Reduction of Order, Nonhomogeneous Equations; Method of Undetermined Coefficients, Variation of Parameters.

Mechanical Vibrations Variation of Parameters; Undamped free vibrations; Damped free vibrations; Critical damping value; Forced vibrations with damping; Transient and SteadyState Solutions; Maximum amplitude and resonance; Undamped Equatiion and Beats.
[VideoLecture] [Lecture Notes]
 Variation of Parameters; Undamped free vibrations; Damped free vibrations; Critical damping value; Forced vibrations with damping; Transient and SteadyState Solutions; Maximum amplitude and resonance; Undamped Equatiion and Beats.

Power Series Methods Review of Power Series; Series Solutions near ordinary points; Euler equations and regular singular points.
[VideoLecture] [Lecture Notes]
 Review of Power Series; Series Solutions near ordinary points; Euler equations and regular singular points.

Series Solutions of ODEs, Bessel Functions Solutions of ODEs near regular singular points; Bessel's Equation.
[VideoLecture] [Lecture Notes]
 Solutions of ODEs near regular singular points; Bessel's Equation.

Laplace Transform and ODEs Laplace Transform; Convolution.
[VideoLecture] [Lecture Notes]
 Laplace Transform; Convolution.

Bessel's Equation Bessel's equation and solutions; Bessel Equation of Order Zero; Bessel Equation of Order OneHalf; Bessel Equation of Order One.
[VideoLecture] [Lecture Notes]
 Bessel's equation and solutions; Bessel Equation of Order Zero; Bessel Equation of Order OneHalf; Bessel Equation of Order One.

Introduction to the Laplace Transform Definition of the Laplace transform; Solution of Initial Value Problems; Examples of Elementary Laplace Transforms; Step Functions.
[VideoLecture] [Lecture Notes]
 Definition of the Laplace transform; Solution of Initial Value Problems; Examples of Elementary Laplace Transforms; Step Functions.

Differential Equations and the Laplace Transform Differential equations with discontinuous forcing terms; Solution smoothness; Impulse Functions and impulse response; The convolution integral, transfer function.
[VideoLecture] [Lecture Notes]
 Differential equations with discontinuous forcing terms; Solution smoothness; Impulse Functions and impulse response; The convolution integral, transfer function.

TwoPoint Boundary Value Problems and Fourier Series Twopoint boundary value problems; Fourier Series; The Fourier Convergence Theorem; Gibbs Phenomenon.
[VideoLecture] [Lecture Notes]
 Twopoint boundary value problems; Fourier Series; The Fourier Convergence Theorem; Gibbs Phenomenon.

The Fourier Convergence Theorem, Separation of Variables and the Heat Diffusion Equation The Fourier Convergence Theorem; Even and Odd Functions; Even and odd periodic extensions of functions; Separation of variables for the solution of hear conduction problems; Eigenvalues, eigenvectors and Fourier expansion; Solving boundary value problems with essential, natural and mixed boundary conditions.
[VideoLecture] [Lecture Notes]
 The Fourier Convergence Theorem; Even and Odd Functions; Even and odd periodic extensions of functions; Separation of variables for the solution of hear conduction problems; Eigenvalues, eigenvectors and Fourier expansion; Solving boundary value problems with essential, natural and mixed boundary conditions.

The Wave Equation and the Laplace Equation Vibrations of an elastic string with a nonzero initial displacement; Fundamental modes; Elastic String with a nonzero initial velocity; Laplace's equation, Dirichlet and Neumann problems; Dirichlet problem in a rectangular and a circle.
[VideoLecture] [Lecture Notes]
 Vibrations of an elastic string with a nonzero initial displacement; Fundamental modes; Elastic String with a nonzero initial velocity; Laplace's equation, Dirichlet and Neumann problems; Dirichlet problem in a rectangular and a circle.

Introduction to SturmLiouville Theory Generalization of twopoint boundary value problems; SturmLiouville Boundary Value Problems; Lagrange's identity; Real eigenvalues and eigenfunctions, orthogonality, orthonornal eigenfunctions, examples; Eigenfunction expansions, convergence theorem; SturmLiouville problems and algebraic eigenvalue problems; Selfadjoint problems.
[VideoLecture] [Lecture Notes]
 Generalization of twopoint boundary value problems; SturmLiouville Boundary Value Problems; Lagrange's identity; Real eigenvalues and eigenfunctions, orthogonality, orthonornal eigenfunctions, examples; Eigenfunction expansions, convergence theorem; SturmLiouville problems and algebraic eigenvalue problems; Selfadjoint problems.

SturmLiouville Theory Eigenfunction expansion, Convergence theorem, Examples; SourmLiouville problems, Algebraic eigenvalue problems and Linear Operator Theory; Selfadjoint problems; Nonhomogeneous boundary value problems, Eigenfunction expansion, Formal solution to the nonhomogeneous problem, Fredholm alternative, examples; Solving the nonhomogeneous transient heat conduction problem, speed of convergence of the solution.
[VideoLecture] [Lecture Notes]
 Eigenfunction expansion, Convergence theorem, Examples; SourmLiouville problems, Algebraic eigenvalue problems and Linear Operator Theory; Selfadjoint problems; Nonhomogeneous boundary value problems, Eigenfunction expansion, Formal solution to the nonhomogeneous problem, Fredholm alternative, examples; Solving the nonhomogeneous transient heat conduction problem, speed of convergence of the solution.

Singular SturmLiouville Problems, Approximation Theory Singular SturmLiouville problems, Examples; Bessel series expansion, Vibrations of a circular elastic membrane; Pointwise and mean convergence, mean square error, approximation theory, square integrable functions.
[VideoLecture] [Lecture Notes]
 Singular SturmLiouville problems, Examples; Bessel series expansion, Vibrations of a circular elastic membrane; Pointwise and mean convergence, mean square error, approximation theory, square integrable functions.

System of ODEs Basic Theory of Systems of FirstOrder Linear Equations; Homogeneous Linear Systems with Constant Coefficients; Complex Eigenvalues.
[VideoLecture] [Lecture Notes]
 Basic Theory of Systems of FirstOrder Linear Equations; Homogeneous Linear Systems with Constant Coefficients; Complex Eigenvalues.

System of ODEs, Phase Plane and Qualitative Methods Linear System of ODEs; Critical Points for Different Cases; Real and Complex Eigenvalues; Repeated Eigenvalues; Solution of Homogeneous and NonHomogeneous Systems of ODEs; Fundamental Matrices.
[VideoLecture] [Lecture Notes]
 Linear System of ODEs; Critical Points for Different Cases; Real and Complex Eigenvalues; Repeated Eigenvalues; Solution of Homogeneous and NonHomogeneous Systems of ODEs; Fundamental Matrices.

Linear ODE Systems and Stability, Nonlinear ODE Systems Criteria for critical points, Stability; Nonlinear systems of ODEs, Linearization and Stability.
[VideoLecture] [Lecture Notes]
 Criteria for critical points, Stability; Nonlinear systems of ODEs, Linearization and Stability.

Nonlinear Differential Equations and Stability The Phase Plane, Linear Systems; Autonomous Systems and Stability; Locally Linear Systems; Competing Species, Predator–Prey Equations.
[VideoLecture] [Lecture Notes]
 The Phase Plane, Linear Systems; Autonomous Systems and Stability; Locally Linear Systems; Competing Species, Predator–Prey Equations.
Homework
 September 1, Homework 1
 Linear Independence, Elimination, LU Factorization, GaussJordan and the Inverse of a Matrix, Nullspace and Column Space
[Homework] [Solution]
 Linear Independence, Elimination, LU Factorization, GaussJordan and the Inverse of a Matrix, Nullspace and Column Space
 September 16, Homework 2
 September 24, Homework 3
 October 2, Homework 4
 Properties of Determinants, Orthogonality, Eigenvalues and Eigenvectors, Diagonalization
[Homework] [Solution]
 Properties of Determinants, Orthogonality, Eigenvalues and Eigenvectors, Diagonalization
 October 8, Homework 5

October 15, Homework 6 
October 29, Homework 7 
November 8, Homework 8 
October 15, Homework 6 
October 29, Homework 7 
November 8, Homework 8 
November 18, Homework 9 
November 26, Homework 10 
Exams
Course info and references
Credit: 3 Units
Lectures: MWF, 10:30  11:20 am, DeBartolo Hall 215.
Professor: Nicholas Zabaras, 311 I Cushing Hall, nzabaras@gmail.com
Teaching Assistants: Zhuogang Peng, zpeng5@nd.edu, Samaresh Midya, smidya@nd.edu
Office hours: Teaching Assistants, Mondays (and if there is demand Wednesdays) 5:15  6:15 p.m. (DeBartolo 203); N. Zabaras, Tuesdays 5:006:00 pm (Cushing 311 I).
Course description: This course provides a rigorous review of linear algebra including vector spaces and subspaces, eigenvalues and eigenvectors, orthogonality, singular value decomposition, linear transformations; Applications of linear algebra to networks, structures, estimation, Lagrange multipliers and optimization will be discussed. Also covered are: differential equations of equilibrium; Laplace's equation and potential flow; boundaryvalue problems and SturmLiuville Theory; system of linear ODEs, nonlinear differential equations and stability; Fourier series; discrete Fourier transform; convolution; and applications.
Goals for the Course: The purpose of the course is to provide you useful insights and understanding of mathematical methods that will be relevant to your research work. While the complexity of calculations will be kept to a minimum, we will put emphasis on integrating concepts from linear algebra to understand the nature of the solutions of systems of linear and nonLinear differential equations. Our goal is to identify the underlying patterns in many applications and allow clear understanding of the nature of the solutions of equations and development of efficient methods to compute them.
Intended audience: Graduate Students in Engineering and the Sciences. Qualified undergraduates can also be enrolled.
References of General Interest: The course lecture slides will become available on the course web site. Please note that the lecture slides are to be read on the internet  and not to be printed. For in depth study, readings from the recommended textbooks will be highlighted. While there is no required text for this course, most of the lectures will follow closely the books by G. Strang and W.E. Boyce and R.C. DiPrima. The books by E. Kreysig and M. Greenberg provide comprehensive material for all lectures (and beyond) and can be useful for subsequent engineering Mathematics courses.
Recommended Textbooks:

 Gilbert Strang, Introduction to Linear Algebra, Fifth Edition 2007.
 William E. Boyce and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 10th Edition.
 Erwin Kreysig, Advanced Engineering Mathematics, 10th Edition.
 Michael Greenberg, Advanced Engineering Mathematics, 2nd Edition.
Homework: Assigned every week. We anticipate approximately ten homework sets. While ideally all homeworks should be submitted typed to allow easy readability, as a minimum you will need to scan your work and submit it electronically. Your homework solutions should be accompanied by any computer programs (e.g. MatLab scripts, data files, Readme files, etc.) and mailed by midnight of the due date tothis Email address. All attachments should arrive on an appropriately named zipped directory (e.g. HW1_Submission_YourName.rar). We will be asking for volunteers weekly to prepare Latex solutions to each homework set. We much appreciate your help.
Exams: There will be three midterm takehome exams (no final). The designated days for the exam are as follows: September 27th, November 2nd and December 6th. The exams are two hrs long but you will be provided freedom within 24 hrs of selecting your start. Managing of downloading the exams and their solution within 2 hrs will be provided through Sakai. Note that we reserve the right to change the format of the exams if it becomes clear that we cannot guarantee the university's honor code.
Grading: Homework 10% and each exam 30%.
Prerequisites: Elementaty Linear Algebra, Differential Calculus, Previous exposure to differential equations. Some programming background in MatLab (or if you desire in Mathematica, Python, etc.) is required for some of the homework. Also it will be helpful to acquire Latex skills in assisting with the homework solution preparation.
Honor Code: Students are expected to understand and abide by the principles and procedures set forth in the University of Notre Dame Academic Code of Honor and uphold the pledge that "As a member of the Notre Dame community, I acknowledge that it is my responsibility to learn and abide by principles of intellectual honesty and academic integrity, and therefore I will not participate in or tolerate academic dishonesty". Students may collaboratively discuss course assignments, but are expected to write and complete their own assignments independently. Downloading solutions from the web or other sources is not allowed. Finally, there should be absolutely no interactions between students regarding the takehome exams.
Syllabus
 Introduction to Vectors and Matrices
 Lengths and Dot Products
 Linear Combinations of Vectors
 Linear Equations
 Elimination using Matrices
 Inverse Matrices, Transposes and Permutations
 A=LU Factorization
 Vector Spaces and Subspaces
 Spaces of Vectors
 The nullspace of A
 Independence, Basis, Dimension
 The Four Fundamental Subspaces
 Orthogonality
 Projections and Least Squares
 Orthonormal Bases and GramSchmidt
 Eigenvalues and Eigenvectors
 Properties of Determinants, Cramer's Rule and Inverses
 Eigenvalue Problem
 Diagonalizing a Matrix
 Symmetric and Positive Definite Matrices
 Applications to Systems of ODEs
 Singular Value Decomposition
 Fundamentals of SVD
 Principal Component Analysis
 Geometry of SVD
 Linear Transformations
 Matrix Representation
 Optimal Basis Representation
 Applications of Linear Algebra
 Hermitian and Unitary Matrices, Fast Fourier Transform
 Linear Algebra for Functions
 Graphs and Networks
 Markov Matrices
 Linear Programming
 FirstOrder Differential Equations
 Integrating Factors
 Numerical Approximations
 Existence and Uniqueness of Solution
 First Order Difference Equations
 SecondOrder Linear Equations
 Solutions of Linear Homogeneous Equations
 Reduction of Order
 Method of Undetermined Coefficient
 Variation of Parameters
 Forced Vibrations
 Series Solutions of 2nd Order Linear Equations
 Power Series Fundamentals
 Series Solutions Near an Ordinary Point
 Euler Equations
 Series Solutions Near a Regular Singular Point, Bessel's Equation
 The Laplace Transform
 Solution of Initial Value Problems
 Differential Equations, Forcing Functions, Impulse Functions
 Convolution
 Systems of First Order Linear Equations
 Homogeneous Linear Systems with Constant Coefficients
 Nonhomogeneous Linear Systems
 Nonlinear Differential Equations and Stability
 The Phase Plane, Linear Systems and Stability
 Locally Linear Systems
 PredatorPrey Equations
 Liapunov's Second Method
 Limit Cycles
 Chaos and Strange Attractors, The Lorenz Equations
 Introduction to Partial Differential Equations
 Fourier Series and The Fourier Convergence Theorem
 Separation of Variables; Heat Conduction in a Rod
 The Wave Equation: Vibrations of an Elastic String
 Laplace’s Equation
 Boundary Value Problems and SturmLiuville Theory
 Sturm–Liouville Boundary Value Problems
 Nonhomogeneous Boundary Value Problems
 Singular Sturm–Liouville Problems
 Bessel Series Expansion
 Series of Orthogonal Functions: Mean Convergence