# Linear Algebra

## Course Overview

Calculation and theory go hand-in-hand in this challenging semester-long course, whose depth and pacing are based on linear algebra courses at the university level. Familiarity with matrices and solving systems of linear equations is assumed as the course quickly moves through more computational topics like matrix algebra and Gauss-Jordan elimination to make time for more theoretical topics like linear transformations and abstract vector spaces before the first exam. In the second half of the course students use the knowledge and skills they have developed to understand real-world applications involving data fitting and dynamical systems. The course ends with a return to theory as we prove the Spectral theorem and explore its consequences. Students are introduced to each topic through pre-class assignments accessed via the LMS. Students are expected to read the textbook, watch associated videos, and answer reading comprehension questions prior to each 90-minute live session (conducted twice a week via video conferencing software). For about half of the class period students work in small groups and tackle challenging problems. The rest of the session is devoted to student presentations, whole group discussion, and answering student questions. Students are encouraged to discuss how they conceptualize each topic, their problem-solving strategies, and the applicability of the ideas to real-world problems. Assessments include a mixture of concrete computations, applications to real-world problems, and proofs of theoretical results. Students are introduced to software that can perform standard matrix computations. However, the majority of such calculations are done by hand, not just because they build character, but because they build intuition.

### Unit 1: Linear Equations and Linear Transformations

This unit develops the theory of linear systems, matrices, and linear transformations. The unit begins with a review of linear systems and methods (algebraic and geometric) of finding solutions to linear systems of arbitrary size. In group work and individual homework assignments, students are asked to prove general results regarding solutions of arbitrary systems and then illustrate those results using Gauss-Jordan elimination when appropriate. Next, students are introduced to three independent definitions of a linear transformation through pre-class readings and instructional videos. Through class discussion and formative in-class assessments during our twice-weekly 90-minute sessions, students explore the relationships between the definitions, ultimately settling on one for the class definition and establishing the others as theorems. A strong focus is placed on the geometric interpretation of linear transformations and matrix algebra.

### Unit 2: Subspaces, Bases, and Dimension

This unit explores the properties of finite-dimensional real vector spaces. The unit begins with an introduction to the notions of the image and kernel of a linear transformation. These two important subspaces are used to guide investigation into more general properties of finite-dimensional real vector spaces such as basis, dimension, coordinates, and subspaces. These investigations take place through a combination of pre-class readings, instructional videos, and in-class discussion during our twice-weekly 90-minute synchronous sessions. In group work and individual homework assignments, students are expected to demonstrate the ability to use concepts from Unit 1 to construct bases for various vector spaces and perform coordinate transformations. In addition, students are asked to prove theoretical results regarding the relationship among new concepts (e.g., the rank-nullity theorem) as well as how these new concepts relate to topics in Unit 1 (e.g., what the dimension of the kernel tells us about uniqueness of solutions to linear systems). The unit ends with a summative exam covering topics in Units 1 and 2.

### Unit 3: Abstract Vector Spaces

This unit develops the theory of abstract vector (linear) spaces. Students are introduced to the formal definition of a vector space (along with basic examples) through pre-class readings and instructional videos. Through in-class discussions and formative assessments students generalize the concepts of Unit 2 into this more abstract setting. This work lays the foundation for understanding the important theoretical concept of an isomorphism and the important practical skill of changing between bases in an abstract vector space.

### Unit 4: Orthogonality and Least Squares

This unit leans heavily into practical applications of the theory of linear algebra. The concepts of orthonormality and orthogonal projection are introduced in the service of developing the Gram-Schmidt process of orthonormalization and the QR factorization of a matrix. These ideas and their geometric interpretations are further developed to provide the theoretical foundation for the method of least squares approximation. With that framework in place, students then perform this data fitting technique on data sets from economics, astronomy, and population biology. Students spend each live session working collaboratively on practice problems, asking questions, and presenting solutions to the entire class.

### Unit 5: Determinants

This brief unit begins by exploring three alternate but equivalent definitions of the determinant. We then shift to the practical matters of calculating determinants and establishing the basic properties of determinants. The unit wraps with a dive into the geometric interpretation of the determinant as a scaling factor and its use in finding areas and volumes. Students spend each live session working collaboratively on practice problems, asking questions, and presenting solutions to the entire class.

### Unit 6: Eigenvalues, Eigenvectors, Symmetric Matrices, and Quadratic Forms

This capstone unit brings together ideas from throughout the semester to establish perhaps the most theoretically important result in all of linear algebra: Hilbert’s Spectral Theorem. Along the way students learn the concepts of eigenvectors and eigenvalues (real and complex) and learn strategies for how to calculate them. Students then apply these ideas to population biology and dynamical systems. Finally, Hilbert’s spectral theorem and its geometric counterpart (the principal axis theorem) are proved in all their glory. Applications of these theorems to signal processing, fluid dynamics, and optimization are briefly explored and the end of the unit. Students spend each live session working collaboratively on practice problems, asking questions, and presenting solutions to the entire class.