Linear Algebra II: Matrices and Linear Transformations from edX

A strong foundation in mathematics is critical for success in all science and engineering disciplines. Whether you want to make a strong start to a master’s degree, prepare for more advanced courses, solidify your knowledge in a professional context or simply brush up on fundamentals, this course will get you up to speed.

In many engineering master’s programs, you need to be familiar with linear algebra. This course will enable you to review the relevant topics.

This course focuses on matrices and linear transformations. Topics covered include matrix algebra, determinants, eigenvalues and eigenvectors, diagonalization and singular value decomposition. The course will help you refresh your knowledge, test your skills and review the relations between the many concepts in linear algebra.

The linear algebra courses within this series will offer you an overview of this branch of mathematics common to most engineering bachelor’s programs. They provide enough depth to cover the linear algebra you need to succeed in your engineering master’s/profession in areas such as computer graphics, systems and control, machine learning, quantum computing and more.

This is a review courseThis self-contained course is modular, so you do not need to follow the entire course if you wish to focus on a particular aspect. As a review course you are expected to have previously studied or be familiar with most of the material. Hence the pace will be higher than in an introductory course.

This format is ideal for refreshing your bachelor level mathematics and letting you practice as much as you want. Through the Grasple platform, you will have access to plenty of exercises and receive intelligent, personal and immediate feedback.

What's inside

Learning objectives

Perform algebraic operations on matrices such as matrix multiplication and matrix inversion.
Recognize linear transformations, apply their properties and find the standard matrix.
Find the determinant of a matrix and apply properties of determinants in the context of algebra and geometry.
Find eigenvalues, eigenvectors and eigenspaces of a matrix.

Diagonalize a matrix if possible and perform other similarity transformations.
Apply properties of symmetric matrices.
Find the singular value decomposition of a matrix.

Perform algebraic operations on matrices such as matrix multiplication and matrix inversion.
Recognize linear transformations, apply their properties and find the standard matrix.
Find the determinant of a matrix and apply properties of determinants in the context of algebra and geometry.
Find eigenvalues, eigenvectors and eigenspaces of a matrix.
Diagonalize a matrix if possible and perform other similarity transformations.
Apply properties of symmetric matrices.
Find the singular value decomposition of a matrix.

Syllabus

Week 1:

matrix multiplication and addition

matrix inversion

Week 2:

linear transformations

standard matrix of a linear transformation

examples of linear transformations in geometry

Week 3:

determinants

methods to find determinants

applications of determinants

Week 4:

eigenvalues and eigenvectors

eigenspaces

characteristic polynomials

complex eigenvalues

multiplicities of eigenvalues

Week 5:

diagonalization

similarity transformations

coordinate transformations

Week 6:

symmetric matrices

quadratic forms

singular value decomposition

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Helps learners refresh their linear algebra skills

Covers a wide range of topics such as matrices, transformations, and decompositions

Taught by qualified and experienced instructors

Provides flexibility with its modular format

Offers immediate feedback with its Grasple platform

Requires learners to have a prior understanding of linear algebra

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 II: Matrices and Linear Transformations with these activities:

Practice Basic Matrix Operations

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Refreshing your skills in basic matrix operations will provide a solid foundation for understanding more advanced concepts.

Browse courses on Matrix Algebra

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Review the basic operations such as addition, subtraction, and multiplication of matrices.
Solve simple matrix equations to practice your skills.
Use online calculators or software to check your answers and identify any areas for improvement.

Review Notes and Course Materials

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By reviewing previous notes and course materials, you can refresh your memory on the fundamental concepts of linear algebra, making it easier to follow the course.

Browse courses on Linear Algebra

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Gather your notes and course materials from previous math courses, particularly those covering linear algebra.
Skim through the materials, paying attention to key concepts such as matrix operations, determinants, and eigenvalues.
Identify areas where you need additional review and focus on those topics.

Watch Video Tutorials

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Watching video tutorials provides a visual and interactive way to learn linear algebra concepts, especially helpful for understanding complex topics.

Browse courses on Linear Algebra

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Find reputable online resources or platforms that offer video tutorials on linear algebra.
Choose tutorials that cover specific topics you need to review or strengthen.
Take notes or summarize the key points presented in the tutorials.
Pause and rewind the videos as needed to fully grasp the concepts.

Five other activities

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Join a Study Group

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Joining a study group provides an opportunity to discuss concepts with peers, clarify doubts, and reinforce your understanding.

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Find or form a study group with classmates who share similar learning goals.
Establish regular meeting times and stick to them.
Take turns presenting concepts, solving problems, and leading discussions.
Encourage active participation and ask questions to deepen your understanding.

Solve Practice Problems

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Solving practice problems will help you solidify your understanding of linear algebra concepts and improve your problem-solving skills.

Browse courses on Matrix Algebra

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Find practice problems from textbooks, online resources, or your instructor.
Set aside dedicated time to work on the problems.
Start with easier problems and gradually move on to more challenging ones.
Check your solutions against provided answers or consult with your instructor for guidance.

Create a Study Guide

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Creating a study guide will help you organize and synthesize the key concepts of linear algebra, making it easier to retain the information.

Browse courses on Linear Algebra

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Identify the main topics and subtopics covered in the course.
Summarize each topic in your own words, including key definitions, theorems, and examples.
Organize the material in a logical and easy-to-follow format.
Review and refine your study guide regularly.

Create a Concept Map

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Creating a concept map helps visualize the relationships and connections between different concepts in linear algebra, improving your overall understanding.

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Start with a central concept or topic.
Identify related concepts and connect them using lines or arrows.
Label the connections with brief explanations or examples.
Expand the concept map by adding more concepts and connections.

Contribute to Open Source Projects

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Contributing to open source projects related to linear algebra can provide practical experience and deepen your understanding.

Browse courses on Linear Algebra

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Identify open source projects on platforms like GitHub that focus on linear algebra.
Explore the codebase and documentation to understand the project's goals and implementation.
Identify areas where you can contribute, such as bug fixes, feature enhancements, or documentation improvements.
Create a pull request to submit your contributions and engage with the project maintainers.

Career center

Learners who complete Linear Algebra II: Matrices and Linear Transformations will develop knowledge and skills that may be useful to these careers:

Quantitative Analyst

Quantitative Analysts, also known as "Quants," are professionals who use mathematical and statistical methods to analyze financial data, identify trends, and develop trading strategies. The course "Linear Algebra II: Matrices and Linear Transformations" provides a strong foundation in linear algebra, which is essential for success in quantitative finance. The course covers topics such as matrix algebra, determinants, eigenvalues and eigenvectors, diagonalization, and singular value decomposition, which are all used extensively in financial modeling and analysis.

See salaries and explore the career path for Quantitative Analyst

Data Scientist

Data Scientists use scientific methods, processes, algorithms, and systems to extract knowledge and insights from data in various forms, both structured and unstructured. The course "Linear Algebra II: Matrices and Linear Transformations" provides a strong foundation in linear algebra, which is essential for data science. The course covers topics such as matrix algebra, determinants, eigenvalues and eigenvectors, diagonalization, and singular value decomposition, which are all used extensively in data analysis and machine learning.

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Machine Learning Engineer

Machine Learning Engineers design, develop, and deploy machine learning models to solve real-world problems. The course "Linear Algebra II: Matrices and Linear Transformations" provides a strong foundation in linear algebra, which is essential for machine learning. The course covers topics such as matrix algebra, determinants, eigenvalues and eigenvectors, diagonalization, and singular value decomposition, which are all used extensively in machine learning algorithms.

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Operations Research Analyst

Operations Research Analysts use mathematical and analytical methods to solve complex problems in business and industry. The course "Linear Algebra II: Matrices and Linear Transformations" provides a strong foundation in linear algebra, which is essential for operations research. The course covers topics such as matrix algebra, determinants, eigenvalues and eigenvectors, diagonalization, and singular value decomposition, which are all used extensively in optimization and decision-making.

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Actuary

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Financial Analyst

Financial Analysts use financial data and analysis to make investment recommendations. The course "Linear Algebra II: Matrices and Linear Transformations" provides a strong foundation in linear algebra, which is essential for financial analysis. The course covers topics such as matrix algebra, determinants, eigenvalues and eigenvectors, diagonalization, and singular value decomposition, which are all used extensively in portfolio management and investment analysis.

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Software Engineer

Software Engineers design, develop, and maintain software systems. The course "Linear Algebra II: Matrices and Linear Transformations" may be useful for Software Engineers who work on graphics, computer vision, or machine learning applications. The course covers topics such as matrix algebra, determinants, eigenvalues and eigenvectors, diagonalization, and singular value decomposition, which are all used extensively in these areas.

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Control Systems Engineer

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Statistician

Statisticians collect, analyze, and interpret data. The course "Linear Algebra II: Matrices and Linear Transformations" may be useful for Statisticians who work on multivariate data analysis. The course covers topics such as matrix algebra, determinants, eigenvalues and eigenvectors, diagonalization, and singular value decomposition, which are all used extensively in multivariate statistics.

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Economist

Economists study the production, distribution, and consumption of goods and services. The course "Linear Algebra II: Matrices and Linear Transformations" may be useful for Economists who work on econometrics. The course covers topics such as matrix algebra, determinants, eigenvalues and eigenvectors, diagonalization, and singular value decomposition, which are all used extensively in econometrics.

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Market Researcher

Market Researchers study consumer behavior and trends. The course "Linear Algebra II: Matrices and Linear Transformations" may be useful for Market Researchers who work on multivariate data analysis. The course covers topics such as matrix algebra, determinants, eigenvalues and eigenvectors, diagonalization, and singular value decomposition, which are all used extensively in multivariate statistics.

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Geophysicist

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Aerospace Engineer

Aerospace Engineers design, develop, and test aircraft, spacecraft, and other aerospace vehicles. The course "Linear Algebra II: Matrices and Linear Transformations" may be useful for Aerospace Engineers who work on flight dynamics and control. The course covers topics such as matrix algebra, determinants, eigenvalues and eigenvectors, diagonalization, and singular value decomposition, which are all used extensively in flight dynamics and control.

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Mechanical Engineer

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Civil Engineer

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