Linear Algebra

Syllabus

Course description

Linear algebra is the study of systems of linear equations (e.g., $ax + by +cz = 0$ where $a,b,c$ are constants and $x,y,z$ are variables). This is naturally equivalent to the study of matrices, with which linear algebra is often equated, which are objects acting on vectors (i.e., objects representing direction that can be added). In this undertaking we will study inner product spaces, which are collections of vectors where we can associate the angle between two vectors (and thus consider orthogonality, or perpendicularity). Finally, we will study determinants which can be thought of as a measure of (signed) volume of collections of $n$ vectors, as well as eigenvalues and eigenvectors which are fundamental objects associated to any linear transformation. Of course, there was a great deal of jargon in this paragraph and we only partially tried to explain it; you will learn its meaning throughout the duration of the course. We will use Elementary Linear Algebra, 8^th^ ed., by Ron Larson.

Course objectives

After completion of the course, students will be able to:

• Understand the object/representation relationship between linear transformations and matrices
• Apply Gaussian and Gauss–Jordan Elimination to solve linear systems
• Compute with matrices (including sum, product, inverse, transpose, determinant)
• Understand vectors (including linear independence, spanning sets, bases, dimension, orthogonality)
• Determine the kernel and range of a linear transformation (and apply the rank-nullity theorem)
• Understand the significance of eigenvalues, eigenvectors and diagonalization

Course Outline And Topics

• Chapter 1: Systems of linear equations

  • Section 1: Introduction to systems of linear equations
  • Section 2: Gaussian Elimination and Guass-Jordan Elimination

• Chapter 2: Matrices

  • Section 1: Operations with matrices
  • Section 2: Properties of matrix operations
  • Section 3: The inverse of a matrix

• Chapter 3: Determinants

  • Section 1: The determinant of a matrix
  • Section 2: Determinants and elementary operations
  • Section 3: Properties of determinants
  • Section 4: Applications of determinants

• Chapter 4: Vector spaces

  • Section 1: Vectors in $\mathbb{R}^n$
  • Section 2: Vectors spaces
  • Section 3: Subspaces of vectors spaces
  • Section 4: Spanning sets and linear independence
  • Section 5: Basis and dimension
  • Section 6: Rank of a matrix and systems of linear equations
  • Section 7: Coordinates and change of basis

• Chapter 5: Inner product spaces

  • Section 1: Length and dot product in $\mathbb{R}^n$
  • Section 2: Inner product spaces
  • Section 3: Orthonormal bases: Gram–Schmidt process

• Chapter 6: Linear transformations

  • Section 1: Introduction to linear transformations
  • Section 2: The kernel and range of a linear transformation
  • Section 3: Matrices for linear transformations
  • Section 4: Transition matrices and similarity
  • Section 5: Applications of linear transformations

• Chapter 7: Eigenvalues and eigenvectors

  • Section 1: Eigenvalues and eigenvectors
  • Section 2: Diagonalization
  • Section 3: Symmetric matrices and orthogonal diagonalization

Grading procedure

• Two tests (20% each)
• WebAssign homework assignments (15% total)
• Application of Linear Algebra Project (15%)
• Comprehensive final exam (25%)
• Attendance and Participation (5%)

Homework will be assigned (almost) every class via WebAssign and will be due by the next class period. Grades will be posted in Blackboard. I will not necessarily announce the WebAssign homework each class, it is your responsibility to know that it should be there (although I will make a specific point to mention it if there will not be homework on any given night).

Notice: attendance is required; for this reason, attendance will be taken each class. Valid reasons to miss class (i.e., excused absences) include, but are not necessarily limited to, documented illness or documented university athletic events, but approval from me is required beforehand.

Important dates

Makeups

If an emergency arises which requires you to miss an exam, I must be made aware at least two hours prior to the start time of your exam. Note: proper documentation will be required before a makeup arrangement is considered.

Academic dishonesty/misconduct

Academic misconduct includes, but is not limited to, cheating, plagiarism and forgery, and soliciting, aiding, abetting, concealing, or attempting such acts. Plagiarism may consist of copying, paraphrasing, or otherwise using written or oral work of another without proper acknowledgment of the source or presenting oral or written material prepared by another as ones own. At minimum, cheating will result in that assignment receiving a grade of zero.

Official communication

Your SIUE student e-mail account is the official method to communicate between you and your instructor. Official communication will not be sent to your personal e-mail (yahoo, wildblue, gmail etc.).

Disability Support Services

If you have a documented disability that requires academic accommodations, please go to Disability Support Services for coordination of your academic accommodations. DSS is located in the Student Success Center, Room 1270; you may contact them to make an appointment by calling (618) 650-3726 or sending an email to disabilitysupport@siue.edu. Additional information is located online at http://siue.edu/dss.

Disclaimer

This syllabus is subject to change by the instructor if deemed necessary for the benefit of student learning or to correct errors and omissions.