Linear Algebra III: Determinants and Eigenvalues from edX

At the beginning of this course we introduce the determinant, which yields two important concepts that you will use in this course. First, you will be able to apply an invertibility criterion for a square matrix that plays a pivotal role in, for example, the understanding of eigenvalues. You will also use the determinant to measure the amount by which a linear transformation changes the area of a region. This idea plays a critical role in computer graphics and in other more advanced courses, such as multivariable calculus.

This course then moves on to eigenvalues and eigenvectors. The goal of this part of the course is to decompose the action of a linear transformation that may be visualized. The main applications described here are to discrete dynamical systems, including Markov chains. However, the basic concepts— eigenvectors and eigenvalues—are useful throughout industry, science, engineering and mathematics.

Prospective students enrolling in this class are encouraged to first complete the linear equations and matrix algebra courses before starting this class.

What you'll learn

Upon completion of this course, learners will be able to:

Compute determinants of using cofactor expansions and properties of determinants
Compute the area of regions in R^3 under a given linear transformation using determinants
Model and solve real-world problems using Markov chains
Verify that a given vector is an eigenvector of a matrix
Verify that a scalar is an eigenvalue of a matrix
Construct an eigenspace for a matrix
Characterize the invertibility of a matrix using determinants and eigenvalues
Apply theorems related to eigenvalues (for example, to characterize the invertibility of a matrix)
Factorize 2 × 2 matrices that have complex eigenvalues
Use eigenvalues to determine identify the rotation and dilation of a linear transform
Apply theorems to characterize matrices with complex eigenvalues
Apply matrix powers and theorems to characterize the long-term behavior of a Markov chain
Construct a transition matrix, a Markov Chain, and a Google Matrix for a given web, and compute the PageRank of the web.

What's inside

Learning objectives

Compute determinants of using cofactor expansions and properties of determinants
Compute the area of regions in r^3 under a given linear transformation using determinants
Model and solve real-world problems using markov chains
Verify that a given vector is an eigenvector of a matrix
Verify that a scalar is an eigenvalue of a matrix
Construct an eigenspace for a matrix
Characterize the invertibility of a matrix using determinants and eigenvalues

Apply theorems related to eigenvalues (for example, to characterize the invertibility of a matrix)
Factorize 2 × 2 matrices that have complex eigenvalues
Use eigenvalues to determine identify the rotation and dilation of a linear transform
Apply theorems to characterize matrices with complex eigenvalues
Apply matrix powers and theorems to characterize the long-term behavior of a markov chain
Construct a transition matrix, a markov chain, and a google matrix for a given web, and compute the pagerank of the web.

Compute determinants of using cofactor expansions and properties of determinants
Compute the area of regions in r^3 under a given linear transformation using determinants
Model and solve real-world problems using markov chains
Verify that a given vector is an eigenvector of a matrix
Verify that a scalar is an eigenvalue of a matrix
Construct an eigenspace for a matrix
Characterize the invertibility of a matrix using determinants and eigenvalues
Apply theorems related to eigenvalues (for example, to characterize the invertibility of a matrix)
Factorize 2 × 2 matrices that have complex eigenvalues
Use eigenvalues to determine identify the rotation and dilation of a linear transform
Apply theorems to characterize matrices with complex eigenvalues
Apply matrix powers and theorems to characterize the long-term behavior of a markov chain
Construct a transition matrix, a markov chain, and a google matrix for a given web, and compute the pagerank of the web.

Traffic lights

Read about what's good

what should give you pause

and possible dealbreakers

Provides analytical tools to understand and solve problems in various disciplines, including computer graphics and multivariable calculus

Focuses on the fundamental concepts of linear algebra, such as determinants and eigenvalues, which form the basis for many higher-level mathematical concepts

Teaches practical applications of linear algebra, such as modeling real-world problems using Markov chains

Covers complex eigenvalues and their applications, providing a deeper understanding of linear transformations

Assumes prior knowledge of linear equations and matrix algebra, which may limit accessibility for beginners

Reviews summary

Deeper dive into determinants and eigenvalues

According to students, this course offers a rigorous and comprehensive exploration of determinants and eigenvalues, fundamental concepts in advanced linear algebra. Learners particularly value the course's emphasis on practical applications, such as modeling with Markov chains and understanding the Google Matrix, which helps to bridge theory with real-world problems. While the content is deemed highly valuable for those pursuing careers in STEM fields, students advise ensuring strong foundational knowledge in preceding linear algebra topics to fully benefit from the challenging yet rewarding material. The course structure, provided by Georgia Tech, ensures a deep, academic understanding.

The course provides a deep, academic dive into the subjects.

"Georgia Tech delivers a truly rigorous academic experience in this course."

"I found the depth of coverage to be excellent for a university-level course."

"It's challenging but provides a thorough understanding that goes beyond the basics."

Excellent examples for applying concepts like Markov chains.

"The sections on Markov chains and PageRank were incredibly helpful for seeing the practical side."

"I appreciate how the course connects abstract linear algebra to real-world problems."

"Learning about the Google Matrix made the abstract concepts feel much more relevant and engaging."

Thorough coverage of determinants and eigenvalues is a strength.

"I found the explanations of determinants and eigenvalues incredibly clear and insightful."

"This course truly deepened my understanding of these fundamental linear algebra topics."

"The detailed approach to each concept was exactly what I needed for advanced study."

Pacing can be fast, requiring diligent self-study for some.

"Sometimes the lectures moved quickly, and I had to rewatch sections frequently."

"While the explanations are clear, the sheer amount of material can be overwhelming."

"I found myself spending a lot of time on extra practice problems to keep up."

Requires a solid foundation in earlier linear algebra concepts.

"I struggled initially because my prior linear algebra knowledge wasn't as strong as needed."

"Make sure you've completed Linear Algebra I and II, as this course builds heavily on them."

"If you're rusty on basic matrix operations, you'll find this course very challenging."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Linear Algebra III: Determinants and Eigenvalues with these activities:

Review matrix algebra

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Review the basics of matrix algebra, including operations such as addition, subtraction, multiplication, and inversion, to prepare for the course.

Browse courses on Matrix Algebra

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Review concepts of matrix addition and subtraction
Review concepts of matrix multiplication
Practice multiplying matrices of different dimensions

Read 'Linear Algebra and Its Applications' by Gilbert Strang

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Expand your understanding of the course material by exploring a comprehensive textbook that provides in-depth explanations, examples, and exercises.

View Linear Algebra for Everyone on Amazon

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Read selected chapters relevant to the course topics
Attempt practice problems to reinforce your understanding
Seek clarification on challenging concepts through online forums

Explore determinant properties

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Follow online tutorials to understand the properties of determinants, such as the product rule, cofactor expansions, and their applications in finding inverses and solving systems of equations.

Browse courses on Determinants

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Watch video tutorials on determinant properties
Complete practice problems on determinant calculations
Apply determinants to solve systems of equations

Five other activities

Expand to see all activities and additional details

Show all eight activities

Solve practice problems on linear transformations

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Engage in regular practice by solving problems related to linear transformations, including matrix multiplication, composition, and geometric interpretations.

Browse courses on Linear Transformations

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Complete practice problems on matrix multiplication and composition
Sketch the transformation of vectors under given linear transformations
Apply linear transformations to solve geometry problems

Practice eigenvalue and eigenvector calculations

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Engage in repetitive exercises to reinforce the concepts of eigenvalues and eigenvectors, including finding eigenvalues and eigenvectors of matrices, and understanding their geometric interpretations.

Browse courses on Eigenvalues

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Calculate eigenvalues and eigenvectors of given matrices
Plot eigenvectors to visualize linear transformations
Apply eigenvalues and eigenvectors to solve real-world problems

Create a visual representation of eigenvalues and eigenvectors

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Deepen your understanding by creating visual representations, such as graphs or animations, to illustrate the concepts of eigenvalues and eigenvectors and their geometric interpretations.

Browse courses on Eigenvalues

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Choose a suitable visualization tool or software
Plot eigenvalues and eigenvectors in a vector space
Animate the transformation of vectors under different eigenvalues

Develop a Markov chain model

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Create a Markov chain model to simulate and analyze a real-world scenario, such as website navigation or disease spread, to gain practical experience in applying Markov chains.

Browse courses on Markov Chains

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Define the states and transition probabilities for the Markov chain
Construct the transition matrix
Analyze the Markov chain to determine steady-state probabilities

Attend a workshop on matrix theory

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Enhance your knowledge and skills by attending a workshop led by industry experts, covering advanced topics and providing opportunities for hands-on practice.

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Research and identify relevant workshops
Register for a workshop that aligns with your learning goals
Actively participate in the workshop activities and discussions

Career center

Learners who complete Linear Algebra III: Determinants and Eigenvalues will develop knowledge and skills that may be useful to these careers:

Data Analyst

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Actuary

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Financial Analyst

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Quantitative Analyst

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Software Engineer

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Computer Engineer

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