Linear Algebra II: Matrix Algebra from edX

Your ability to apply the concepts that we introduced in our previous course is enhanced when you can perform algebraic operations with matrices. At the start of this class, you will see how we can apply the Invertible Matrix Theorem to describe how a square matrix might be used to solve linear equations. This theorem is a fundamental role in linear algebra, as it synthesizes many of the concepts introduced in the first course into one succinct concept.

You will then explore theorems and algorithms that will allow you to apply linear algebra in ways that involve two or more matrices. You will examine partitioned matrices and matrix factorizations, which appear in most modern uses of linear algebra. You will also explore two applications of matrix algebra, to economics and to computer graphics.

Students taking this class are encouraged to first complete the first course in this series, linear equations.

What you'll learn

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

Apply matrix algebra, the matrix transpose, and the zero and identity matrices, to solve and analyze matrix equations.
Apply the formal definition of an inverse, and its algebraic properties, to solve and analyze linear systems.
Characterize the invertibility of a matrix using the Invertible Matrix Theorem.
Apply partitioned matrices to solve problems regarding matrix invertibility and matrix multiplication.
Compute an LU factorization of a matrix and apply the LU factorization to solve systems of equations.
Apply matrix algebra and inverses to solve and analyze Leontif Input-Output problems.
Construct transformation matrices to represent composite transforms in 2D and 3D using homogeneous coordinates.
Construct a basis for a subspace.
Calculate the coordinates of a vector in a given basis.
Characterize a matrix using the concepts of rank, column space, and null space.
Apply the Rank, Basis, and Matrix Invertibility theorems to describe matrices, subspaces, and systems.

What's inside

Learning objectives

Apply matrix algebra, the matrix transpose, and the zero and identity matrices, to solve and analyze matrix equations.
Apply the formal definition of an inverse, and its algebraic properties, to solve and analyze linear systems.
Characterize the invertibility of a matrix using the invertible matrix theorem.
Apply partitioned matrices to solve problems regarding matrix invertibility and matrix multiplication.
Compute an lu factorization of a matrix and apply the lu factorization to solve systems of equations.
Apply matrix algebra and inverses to solve and analyze leontif input-output problems.

Construct transformation matrices to represent composite transforms in 2d and 3d using homogeneous coordinates.
Construct a basis for a subspace.
Calculate the coordinates of a vector in a given basis.
Characterize a matrix using the concepts of rank, column space, and null space.
Apply the rank, basis, and matrix invertibility theorems to describe matrices, subspaces, and systems.

Apply matrix algebra, the matrix transpose, and the zero and identity matrices, to solve and analyze matrix equations.
Apply the formal definition of an inverse, and its algebraic properties, to solve and analyze linear systems.
Characterize the invertibility of a matrix using the invertible matrix theorem.
Apply partitioned matrices to solve problems regarding matrix invertibility and matrix multiplication.
Compute an lu factorization of a matrix and apply the lu factorization to solve systems of equations.
Apply matrix algebra and inverses to solve and analyze leontif input-output problems.
Construct transformation matrices to represent composite transforms in 2d and 3d using homogeneous coordinates.
Construct a basis for a subspace.
Calculate the coordinates of a vector in a given basis.
Characterize a matrix using the concepts of rank, column space, and null space.
Apply the rank, basis, and matrix invertibility theorems to describe matrices, subspaces, and systems.

Traffic lights

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what should give you pause

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Explores theorems and algorithms that apply linear algebra in ways involving two or more matrices

Taught by Greg Mayer, who is recognized for their work in linear algebra

Develops key skills in matrix algebra and inverses, which are core skills for advanced mathematics and computer science

Recommended for students with prior experience in linear equations

Covers the Invertible Matrix Theorem, a fundamental concept in linear algebra

Builds a strong foundation for intermediate learners in linear algebra

Reviews summary

Comprehensive matrix algebra & applications

According to students, Linear Algebra II: Matrix Algebra from Georgia Tech is highly praised for its clear and concise instruction, particularly the instructor's ability to simplify complex topics. The course's well-structured content and logical progression are frequently highlighted. A major strength is its strong focus on practical applications, including valuable sections on Leontief models and computer graphics, which learners found highly relevant for careers. While it demands a solid prerequisite understanding from Linear Algebra I, recent feedback indicates significant improvements in quiz clarity and feedback, addressing prior concerns. Some learners mentioned inconsistent forum support as an area for improvement.

Requires strong mathematical prerequisite knowledge.

"While the course is excellent, it absolutely requires a solid grasp of Linear Algebra I. Without that foundation, some parts felt very fast-paced."

"The content is rigorous and demands significant effort. If you don't have a strong math background, be prepared to spend extra time studying."

"I found some sections, like transformation matrices, to be particularly dense, necessitating thorough review of prior material."

Updates have enhanced assessment clarity and feedback.

"The quiz questions were often poorly worded and the feedback was minimal. It was frustrating trying to figure out why an answer was wrong..."

"They've clearly updated the quizzes and homework. The explanations for incorrect answers are much better now, and the clarity of questions has improved."

"I appreciate that they seem to have revised the assessments; it's a much smoother experience now with clearer expectations."

Connects theoretical knowledge to practical scenarios.

"The sections on Leontief Input-Output models and computer graphics transformations were incredibly useful; it's great to see real-world applications."

"I enrolled to enhance my skills for data science, and this course delivered. The concepts were explained with an eye towards practical use."

"It's helpful to see how matrix algebra applies to economics and computer graphics; it made the learning stick beyond just abstract theory."

Expertly explains complex linear algebra concepts.

"The instructor's explanations were incredibly clear and concise, making complex topics like LU factorization much more digestible for me."

"Professor S. is an exceptional educator; his ability to break down abstract linear algebra into understandable chunks is outstanding."

"I found the course material very well-structured. The lecturer explains the topics in a clear and logical manner, always linking back."

Inconsistent responsiveness in discussion forums.

"The forum support was inconsistent. While some questions were answered quickly, others lingered for days, which was problematic when I was stuck."

"I wish there was more active TA presence in the discussion forums for quicker help with challenging problems."

"Sometimes, when I needed urgent clarification on an assignment, I found the forum responses to be delayed, impacting my progress."

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: Matrix Algebra with these activities:

Review Matrices and Matrix Operations

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Review basic concepts like matrix addition, subtraction, scalar multiplication, and matrix multiplication to strengthen your foundation for more advanced topics.

Browse courses on Matrices

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Recall basic matrix properties and definitions.
Practice performing basic matrix operations on small examples.
Solve simple linear equations using matrix operations.

Participate in a Study Group for Matrix Algebra

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Engage in collaborative learning by joining a study group with peers. Discuss concepts, solve problems together, and reinforce your understanding through peer-to-peer interactions.

Browse courses on Matrix Algebra

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Join or form a study group with other students taking the course.
Schedule regular meetings to discuss course material, solve problems, and quiz each other.

Explore MIT OpenCourseWare Lectures on Matrix Algebra

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Gain additional insights and perspectives by watching video lectures from MIT OpenCourseWare. These lectures provide clear and engaging explanations of matrix algebra concepts.

Browse courses on Matrix Algebra

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Watch lectures on topics relevant to the course, such as matrix operations, inverses, and systems of equations.
Take notes and pause the videos to reflect on the concepts.

Four other activities

Expand to see all activities and additional details

Show all seven activities

Solve Matrix Equations Involving Inverses

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Strengthen your understanding of matrix inverses and their applications by solving various matrix equations involving inverses.

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Review the concept of matrix inverses and their properties.
Practice finding inverses of matrices using various methods.
Solve matrix equations by applying matrix inverses.

Create Visual Aids for Matrix Concepts

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Enhance your comprehension by creating visual aids such as diagrams, charts, or mind maps that represent matrix concepts. This will help you visualize and understand the relationships between matrix elements.

Browse courses on Visual Aids

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Identify key matrix concepts that you want to represent visually.
Choose appropriate visual formats, such as diagrams, charts, or mind maps.
Create visual aids that clearly illustrate the concepts and their relationships.

Read 'Linear Algebra and Its Applications' by Gilbert Strang

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Delve deeper into the concepts covered in the course by reading this classic textbook. It provides comprehensive explanations and insightful examples that will enhance your understanding and retention.

View Linear Algebra for Everyone on Amazon

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Read the relevant chapters covering matrix algebra and linear systems.
Work through the practice problems and exercises provided in the book.

Develop a Matrix Algebra Toolkit

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Create a collection of custom functions or a software tool that automates matrix operations and solves matrix equations. This will provide you with a valuable resource for future work involving matrix algebra.

Browse courses on Custom Functions

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Identify the specific matrix operations and equations that you want to automate.
Design and develop custom functions or a software tool that implements these operations and equations.
Test and refine your tool to ensure accuracy and efficiency.

Career center

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

Quantitative Analyst

Quantitative Analysts use their knowledge of mathematics and computer science to analyze data and identify trends. In this role, you will use matrix algebra and other mathematical concepts to develop and implement quantitative models. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the quantitative models that you will use on a daily basis. Linear Algebra II: Matrix Algebra can help you to succeed as a Quantitative Analyst by providing you with the mathematical foundation you need to excel in this role.

See salaries and explore the career path for Quantitative Analyst

Data Scientist

Data Scientists use their knowledge of mathematics, statistics, and computer science to analyze data and extract insights. In this role, you will use matrix algebra and other mathematical concepts to develop data mining algorithms and models. This course will help you to develop the mathematical skills you need to become a successful Data Scientist. Linear Algebra II: Matrix Algebra will provide you with the mathematical foundation you need to excel in this role.

See salaries and explore the career path for Data Scientist

Operations Research Analyst

Operations Research Analysts use their knowledge of mathematics and computer science to solve business problems. In this role, you will use matrix algebra and other mathematical concepts to develop and implement optimization models. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the optimization models that you will use on a daily basis. Linear Algebra II: Matrix Algebra will help you to develop the mathematical skills you need to become a successful Operations Research Analyst.

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Computer Programmer

Computer Programmers use their knowledge of computer science to develop and implement software applications. In this role, you will use matrix algebra and other mathematical concepts to develop algorithms and data structures. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the algorithms and data structures that you will use on a daily basis. Linear Algebra II: Matrix Algebra will help you to develop the mathematical skills you need to become a successful Computer Programmer.

See salaries and explore the career path for Computer Programmer

Financial Analyst

Financial Analysts use their knowledge of mathematics and economics to analyze financial data and make investment recommendations. In this role, you will use matrix algebra and other mathematical concepts to develop financial models. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the financial models that you will use on a daily basis. Linear Algebra II: Matrix Algebra will provide you with the mathematical foundation you need to excel in this role.

See salaries and explore the career path for Financial Analyst

Actuary

Actuaries use their knowledge of mathematics and statistics to assess risk and develop insurance policies. In this role, you will use matrix algebra and other mathematical concepts to develop actuarial models. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the actuarial models that you will use on a daily basis. Linear Algebra II: Matrix Algebra will provide you with the mathematical foundation you need to excel in this role.

See salaries and explore the career path for Actuary

Mathematician

Mathematicians use their knowledge of mathematics to solve problems and develop new theories. In this role, you will use matrix algebra and other mathematical concepts to conduct research in various fields of mathematics. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the research that you will conduct on a daily basis. Linear Algebra II: Matrix Algebra will help you to develop the mathematical skills you need to become a successful Mathematician.

See salaries and explore the career path for Mathematician

Statistician

Statisticians use their knowledge of mathematics and statistics to collect and analyze data. In this role, you will use matrix algebra and other mathematical concepts to develop statistical models. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the statistical models that you will use on a daily basis. Linear Algebra II: Matrix Algebra will help you to develop the mathematical skills you need to become a successful Statistician.

See salaries and explore the career path for Statistician

Computer Scientist

Computer Scientists use their knowledge of computer science to design and develop software applications. In this role, you will use matrix algebra and other mathematical concepts to develop algorithms and data structures. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the algorithms and data structures that you will use on a daily basis. Linear Algebra II: Matrix Algebra is a key course for anyone who wants to enter the field of Computer Science. It will provide you with the mathematical foundation you need to excel in this role.

See salaries and explore the career path for Computer Scientist

Software Engineer

Software Engineers use their knowledge of computer science to design and develop software applications. In this role, you will use matrix algebra and other mathematical concepts to develop algorithms and data structures. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the algorithms and data structures that you will use on a daily basis. Linear Algebra II: Matrix Algebra is a key course for anyone who wants to enter the field of Software Engineering. It will provide you with the mathematical foundation you need to excel in this role.

See salaries and explore the career path for Software Engineer

Data Analyst

Data Analysts use their knowledge of mathematics and computer science to analyze data and extract insights. In this role, you will use matrix algebra and other mathematical concepts to develop data mining algorithms and models. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the data mining algorithms and models that you will use on a daily basis. Linear Algebra II: Matrix Algebra is a key course for anyone who wants to enter the field of Data Analytics. It will provide you with the mathematical foundation you need to excel in this role.

See salaries and explore the career path for Data Analyst

Economist

Economists use their knowledge of economics to analyze economic data and develop economic models. In this role, you will use matrix algebra and other mathematical concepts to develop economic models. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the economic models that you will use on a daily basis. Linear Algebra II: Matrix Algebra is a key course for anyone who wants to enter the field of Economics. It will provide you with the mathematical foundation you need to excel in this role.

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Financial Manager

Financial Managers use their knowledge of finance to manage the financial resources of a company or organization. In this role, you will use matrix algebra and other mathematical concepts to develop financial models. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the financial models that you will use on a daily basis. Linear Algebra II: Matrix Algebra is a key course for anyone who wants to enter the field of Financial Management. It will provide you with the mathematical foundation you need to excel in this role.

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Market Researcher

Market Researchers use their knowledge of marketing and research to conduct market research studies. In this role, you will use matrix algebra and other mathematical concepts to analyze market research data. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the market research data that you will use on a daily basis. Linear Algebra II: Matrix Algebra is a key course for anyone who wants to enter the field of Market Research. It will provide you with the mathematical foundation you need to excel in this role.

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Operations Manager

Operations Managers use their knowledge of operations management to manage the operations of a company or organization. In this role, you will use matrix algebra and other mathematical concepts to develop operations management models. This course will introduce you to the fundamental concepts of matrix algebra, which will be essential for you to understand the operations management models that you will use on a daily basis. Linear Algebra II: Matrix Algebra is a key course for anyone who wants to enter the field of Operations Management. It will provide you with the mathematical foundation you need to excel in this role.

See salaries and explore the career path for Operations Manager