Applied Linear Algebra: Course Overview

Applied Linear Algebra

Course Description

This is an undergraduate course that teaches both basic theory and applications of linear algebra. Some of the important topics in this course include: systems of linear equations, vector and matrix operations, vector spaces, linear transformation, determinants, eigenvalues, and eigenvectors.

As with most courses, the subject material and pace of the course are often challenging for students enrolled in the class. To help my students I developed my own set of example video problems to assist them with their homework and provide them more examples to study. This collection of example problems is organized by problem type below.

The comprehensive set of videos listed below cover all topics in the course; 92 videos totaling 9+ hours of content.

The textbook used for the course is, "Linear Algebra and Its Applications", 4th edition by David C. Lay.

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Example Problems

Equations and Matrices (8 videos, ~50 minutes)
- Topics include: intersecting lines, solving systems of equations, augmented matrices, reduced row echelon form of a matrix, homogeneous systems of equations, and general solutions of augmented matrices.

Vectors (8 videos, ~39 minutes)
- Topics include: Basic vector computations, linear combinations of vectors, linear independence, and the span of a set of vectors

Linear Transformations (8 videos, ~48 minutes)
- Topics include: Matrix representation of a linear transformation, rotation/reflection of a linear transformation, one-to-one linear transformations, and onto linear transformations.

Matrices (12 videos, ~66 minutes)
- Topics include: Matrix multiplication, inverses, determinants, determinant properties, and Cramer's rule.

Vector Spaces (20 videos, ~127 minutes)
- Topics include: Definition of a vector space and subspace, the null space and row space of a matrix, finding the basis for a vector space, dimension of a subspace, coordinate system representations, and change-of-coordinates matrices.

Eigenvalue and Eigenvectors (9 videos, ~66 minutes)
- Topics include: Finding eigenvalues and eigenvectors of a matrix, computing matrices raised to a power, matrix diagonalization.

Orthogonality and Least Squares Part 1: (13 videos, ~78 minutes)
- Topics include: Inner product, vector norm, dot product, orthogonality, orthogonal complements, projections, and orthogonal basis.

Orthogonality and Least Squares Part 2: (13 videos, ~90 minutes)
- Topics include: Orthonormal sets, orthogonal matrices, the Orthogonal Decomposition Theorem, the Gram-Schmidt algorithm, QR factorization, and least-squares approximations.

Applied Linear Algebra

Example Problems

Equations and Matrices (8 videos, ~50 minutes)

Topics include: intersecting lines, solving systems of equations, augmented matrices, reduced row echelon form of a matrix, homogeneous systems of equations, and general solutions of augmented matrices.

Vectors (8 videos, ~39 minutes)

Topics include: Basic vector computations, linear combinations of vectors, linear independence, and the span of a set of vectors

Linear Transformations (8 videos, ~48 minutes)

Topics include: Matrix representation of a linear transformation, rotation/reflection of a linear transformation, one-to-one linear transformations, and onto linear transformations.

Matrices (12 videos, ~66 minutes)

Topics include: Matrix multiplication, inverses, determinants, determinant properties, and Cramer's rule.

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Vector Spaces (20 videos, ~127 minutes)

Topics include: Definition of a vector space and subspace, the null space and row space of a matrix, finding the basis for a vector space, dimension of a subspace, coordinate system representations, and change-of-coordinates matrices.

​

Eigenvalue and Eigenvectors (9 videos, ~66 minutes)

Topics include: Finding eigenvalues and eigenvectors of a matrix, computing matrices raised to a power, matrix diagonalization.

​

Orthogonality and Least Squares Part 1: (13 videos, ~78 minutes)

Topics include: Inner product, vector norm, dot product, orthogonality, orthogonal complements, projections, and orthogonal basis.

​

Orthogonality and Least Squares Part 2: (13 videos, ~90 minutes)

Topics include: Orthonormal sets, orthogonal matrices, the Orthogonal Decomposition Theorem, the Gram-Schmidt algorithm, QR factorization, and least-squares approximations.

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