General Information:
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- Instructor(s) Name: Paul Tsopméné
- Instructor(s) Email: paul.tsopmene@ubc.ca
- Instructor(s) Office: SCI 259
- Duration: Term 1 Winter 2022 (Sep-Dec 2022)
- Delivery Modality: In-Person
- Course Location: ASC 130
- Course Days: Tue and Thu
- Class Hours: 2:00 PM – 3:30 PM
- Office Hours: Tuesday and Thursday: 10:30 am – 12:00 pm. I am also available by email or appointment if these times do not work for you.
- Course Website: Course materials are available on Canvas. My primary method of communication is through Canvas messages/emails. Make sure you check this website regularly.
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Course Objectives: The main objective of this course is to give an introduction to matrix algebra. Students will be able to understand the fundamental concepts and computations of matrix algebra and use them to solve a wide range of problems. Students will also be exposed to some theoretical aspects of matrix algebra and able to link various concepts/statements mentioned in the course.
Course Overview: This course is an introduction to linear algebra, which is a branch of mathematics that studies systems of linear equations and the properties of matrices (plural of matrix). The concepts of linear algebra are extremely useful in many areas, including physics, economics and social sciences, natural sciences, computer science, data science, and engineering. Due to its broad range of applications, linear algebra is one of the most widely taught subjects in college-level mathematics. In this course we will concentrate on the methods of linear algebra and students will be exposed to some formal proofs. We will become competent in solving systems of linear equations, performing matrix operations, calculating determinants, and finding eigenvalues and eigenvectors. On the theoretical side, we will become comfortable with the vector space R^n and come to understand a matrix as a linear transformation. On the application side, we will see how to use linear systems to solve real-world problems. We will also see how to use determinants to find volumes and eigenvalues and eigenvectors to study/understand some natural phenomena (if time permits).
Contents: Topics include
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- Linear Equations in Linear Algebra: Systems of Linear Equations, Row Reduction and Echelon Forms, Vector Equations, Matrix Equations, Solution Sets of Linear Equations, Linear Independence, Linear Transformations.
- Matrix Algebra: Matrix Operations, The Inverse of a Matrix, Characterizations of Invertible Matrices, Subspaces of , Column Space and Null Space, Basis, Dimension and Rank.
- Determinants: Introduction to Determinants, Properties of Determinants, Applications of Determinants (if time permits).
- Eigenvalues and Eigenvectors: Eigenvectors and Eigenvalues, The Characteristic Equation, Diagonalization and Applications.
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Learning Outcomes: After completing this course, students will be able to:
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- Solve systems of linear equations by using matrices (that is, by row reducing the augmented matrix to an echelon form). If a system has at least one solution, students should be able to write the set of solutions in (parametric) vector form.
- Express a linear system as a vector equation as well as a matrix equation.
- Understand the concepts of linear combination and span. In particular, students should be able to show that a given vector is in the span of a set of vectors.
- Understand the concept of linear independence. In particular, students should be able to determine whether a set of vectors is linearly independent or not.
- Find a linear dependence relation among vectors which are not linearly independent.
- Show that a transformation is linear.
- Show that a transformation is not linear.
- Find the image of a vector under a linear transformation.
- Find the standard matrix of a (geometric) linear transformation.
- Understand the concepts of onto and one-to-one. In particular, students should be able to determine whether a linear transformation is onto or one-to-one or neither.
- Understand a matrix as a linear transformation.
- Perform matrix operations – multiply a matrix by a number, add two matrices, subtract two matrices, multiply two matrices, find the transpose of a matrix.
- Determine whether a matrix is invertible or not. If a matrix is invertible, students should be able to find its inverse.
- Use the properties of transposes and matrix inverses to solve matrix equations.
- Use the Invertible Matrix Theorem to prove statements.
- Determine whether a vector is in the column space or the null space of a matrix.
- Find a basis and the dimension of the column space and the null space of a matrix.
- Use the Rank-Nullity Theorem to find the rank or the dimension of the null space of a matrix.
- Compute the determinant of a matrix using various techniques.
- Find the eigenvalues of a matrix.
- Find the eigenvectors of a matrix – find a basis and the dimension of the eigenspace corresponding to an eigenvalue.
- Determine whether a matrix is diagonalizable or not.
- Compute the nth power of a (diagonalizable) matrix for an arbitrary nonnegative integer n.
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Learning Resources:
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- Textbook (Optional): Linear Algebra and Its Applications, by David C. Lay, Steven R. Lay and Judi J. McDonald, 5th edition, Pearson. This is optional, as the exercise book below is enough.
- Exercise Book: To help students learn the material more efficiently, I created an exercise book for the course with a wide variety of problems, complete step-by-step solutions, and concept summaries. The exercise book is self-contained and free. It is the only required learning resource for this course. Here is the exercise book.
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Course Outline: More information about this course can be found in the course outline.