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.

Good to know

Know what's good

, what to watch for

, 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

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

Show steps

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

Show steps

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

Show steps

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

Show steps

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

Show steps

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

Show steps

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

Show steps

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

Browse courses on Linear Transformations

Show steps

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

Show steps

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

Show steps

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

Show steps

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

Show steps

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

Show steps

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

Show steps

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

Show steps

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

Show steps

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:

Investment Analyst

Investment Analysts may leverage this course. It can give you the skills to compute determinants of matrices using the methods and theorems provided.

See salaries and explore the career path for Investment Analyst

Operations Research Analyst

Operations Research Analysts may leverage this course. It can give you the skills to compute determinants of matrices using the methods and theorems provided.

See salaries and explore the career path for Operations Research Analyst

Statistician

Statisticians may leverage this course. It can give you the skills to compute determinants of matrices using the provided methods and theorems.

See salaries and explore the career path for Statistician

Data Analyst

Data Analysts may leverage this course. It can give you the skills to compute determinants of matrices using the provided methods and theorems.

See salaries and explore the career path for Data Analyst

Financial Analyst

Financial Analysts may leverage this course. It can give you the skills to compute determinants of matrices using the provided methods and theorems.

See salaries and explore the career path for Financial Analyst

Quantitative Analyst

Quantitative Analysts may leverage this course. It can give you the skills to compute determinants of matrices using the methods and theorems provided.

See salaries and explore the career path for Quantitative Analyst

Actuary

Actuaries may leverage this course. It can give you the skills to compute determinants of matrices using the methods and theorems provided.

See salaries and explore the career path for Actuary

Economist

Economists may leverage this course. It can give you the skills to compute determinants of matrices using the methods and theorems provided.

See salaries and explore the career path for Economist

Data Scientist

Data Scientists may leverage this course. It can give you the skills to compute determinants of matrices using the provided methods and theorems. This is a useful skill as it has applications in machine learning, which is closely related to data science.

See salaries and explore the career path for Data Scientist

Chemical Engineer

Chemical Engineers may leverage this course. It covers computer graphics and other advanced courses. These subjects may be used in some chemical engineering work.

See salaries and explore the career path for Chemical Engineer

Software Engineer

Software Engineers may leverage this course. It covers computer graphics and other advanced courses. These subjects relate to software engineering.

See salaries and explore the career path for Software Engineer

Mechanical Engineer

Mechanical Engineers may leverage this course. It covers computer graphics and other advanced courses. These subjects may be used in some mechanical engineering work.

See salaries and explore the career path for Mechanical Engineer

Aerospace Engineer

Aerospace Engineers may leverage this course. It covers computer graphics and other advanced courses. These subjects may be used in some aerospace engineering work.

See salaries and explore the career path for Aerospace Engineer

Electrical Engineer

Electrical Engineers may leverage this course. It covers computer graphics and other advanced courses. These subjects may be used in some electrical engineering work.

See salaries and explore the career path for Electrical Engineer

Computer Engineer

Computer Engineers may leverage this course. It covers computer graphics and other advanced courses. These subjects may be used in some computer engineering work.

See salaries and explore the career path for Computer Engineer