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Linear Algebra

Linear Systems and Matrix Equations

Joseph W. Cutrone, PhD

This is the first course of a three course specialization that introduces the students to the concepts of linear algebra, one of the most important and basic areas of mathematics, with many real-life applications. This foundational material provides both theory and applications for topics in

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This is the first course of a three course specialization that introduces the students to the concepts of linear algebra, one of the most important and basic areas of mathematics, with many real-life applications. This foundational material provides both theory and applications for topics in

mathematics, engineering and the sciences. The course content focuses on linear equations, matrix methods, analytical geometry and linear transformations. As well as mastering techniques, students will be exposed to the more abstract ideas of linear algebra. Lectures, readings, quizzes, and a project all help students to master course content and and learn to read, write, and even correct

mathematical proofs. At the end of the course, students will be fluent in the language of linear algebra, learning new definitions and theorems along with examples and counterexamples. Students will also learn to employ techniques to classify and solve linear systems of equations. This course prepares students to continue their study of linear transformations with the next course in the specialization. .

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What's inside

Syllabus

Introduction to Matrices
In this module we introduce two fundamental objects of study: linear systems and the matrices that model them. We ask two fundamental questions about linear systems, specifically, does a solution exist and if there is a solution, is it unique. To answer these questions, a fundamental invariant needs to be found. We will use the Row Reduction Algorithm Algorithm to see the number of pivot positions in a matrix. These foundational concepts of matrices and row reduction will be revisited over and over again throughout the course so pay attention to new vocabulary, the technical skills presented, and the theory of why these algorithms are performed.
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Traffic lights

Read about what's good
what should give you pause
and possible dealbreakers
Develops fluency in the language of linear algebra, which is highly relevant in math, engineering, and the sciences
Introduces the fundamental concepts of linear equations, matrix methods, analytical geometry, and linear transformations
Provides a strong foundation for those interested in pursuing advanced study in linear transformations
Emphasizes both theory and applications, providing learners with valuable skills and understanding
Taught by Joseph W. Cutrone, PhD, an expert in the field of linear algebra
Requires a prior understanding of algebra

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Reviews summary

Foundational linear algebra for stem

According to learners, this course offers a strong foundation in linear algebra, particularly for those in STEM fields. Students consistently praise the clear and concise lectures and the instructor's ability to break down difficult concepts, making complex topics like row reduction and matrix operations accessible. The course is noted for its balanced approach to theory and application, providing practical relevance. However, some students find the pace too fast for abstract ideas and proofs, potentially requiring prior mathematical maturity or supplementary resources. A few reviewers also suggested that the course could benefit from more practice problems or examples to solidify understanding.
This course is the first in a multi-part linear algebra series.
"It's a great stepping stone for the rest of the specialization."
"This course provided a good foundation for subsequent courses."
Students desire additional practice problems and examples.
"I wish there were more practice problems or examples to solidify the understanding."
"More examples on complex problems would be beneficial."
"I think it needs more exercises."
Effectively integrates theoretical concepts with practical uses.
"I appreciated the focus on both the theoretical aspects and the practical applications of linear algebra."
"The emphasis on matrix methods and solving linear systems was directly applicable to my studies."
"The focus on both theory and application is balanced, making it relevant for various fields."
Provides essential building blocks for further study in linear algebra.
"This course is an excellent introduction to Linear Algebra for anyone needing a strong foundation."
"It truly built a solid base for understanding vectors and linear transformations."
"This course clarified so many concepts I struggled with in traditional textbooks."
"I gained a solid foundation from completing this course, especially on linear systems and matrix equations."
Instructor excels at simplifying complex linear algebra concepts.
"The lectures are clear, concise, and the pace is perfect for understanding complex topics."
"The instructor breaks down difficult concepts into manageable pieces."
"I found the explanations of row reduction and matrix operations particularly helpful."
"The explanations are incredibly clear, and the way concepts build upon each other is very effective."
Pacing and depth of abstract topics can be difficult for some learners.
"I struggled with the proofs and definitions and found myself frequently re-watching lectures."
"Some of the proofs were a bit challenging to follow, especially without more detailed step-by-step derivations."
"I felt it moved a bit too fast through some of the abstract ideas, especially linear transformations."
"I found this course quite difficult and overwhelming. The pace was too fast for me, and I felt lost early on."

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: Linear Systems and Matrix Equations with these activities:
Review Precalculus Concepts
Reinforce your foundational knowledge in precalculus to strengthen your understanding of linear algebra concepts.
Browse courses on Precalculus
Show steps
  • Review the unit circle and trigonometric identities.
  • Practice solving systems of equations.
  • Familiarize yourself with matrices and their operations.
Solve Linear Equation Systems Practice
Improve your problem-solving skills and reinforce your understanding of solving linear equations by engaging in dedicated practice.
Browse courses on Linear Equations
Show steps
  • Find practice problems online or in textbooks.
  • Start with easier problems and gradually increase the complexity.
  • Check your answers and identify areas for improvement.
Work through Linear Algebra Video Tutorials
Supplement your learning by exploring online video tutorials that provide clear explanations and visual demonstrations of linear algebra concepts.
Show steps
  • Search for reputable online sources that offer video tutorials on linear algebra.
  • Choose tutorials that align with your current understanding and gradually increase the difficulty.
  • Take notes and pause the videos to fully grasp the concepts.
Three other activities
Expand to see all activities and additional details
Show all six activities
Join a Study Group for Linear Algebra
Collaborate with peers to enhance your understanding of linear algebra through discussions, problem-solving, and knowledge sharing.
Show steps
  • Connect with classmates or fellow students interested in forming a study group.
  • Establish regular meeting times and a designated study space.
  • Take turns presenting concepts, leading discussions, and working through problems together.
Read 'Linear Algebra and Its Applications' by David C. Lay
Expand your knowledge and gain a deeper understanding of linear algebra by delving into a comprehensive textbook that provides a thorough exploration of the subject.
View Linear Algebra and Its Applications, Global... on Amazon
Show steps
  • Obtain a copy of the book.
  • Allocate dedicated time for reading and studying.
  • Take notes, highlight key concepts, and solve practice problems to reinforce your learning.
Create a Visual Representation of Linear Transformations
Demonstrate your understanding by visually representing linear transformations using diagrams, graphs, or animations, which can enhance your comprehension and ability to explain the concept to others.
Show steps
  • Choose a linear transformation to represent.
  • Decide on the appropriate visual representation (e.g., diagram, graph, animation).
  • Create the visual representation using software or by hand.
  • Present your visual representation to your peers or instructor for feedback.

Career center

Learners who complete Linear Algebra: Linear Systems and Matrix Equations will develop knowledge and skills that may be useful to these careers:
Quantitative Analyst
A Quantitative Analyst uses mathematical and statistical techniques to analyze financial data and make investment decisions. This course may be useful for a Quantitative Analyst by strengthening your understanding of linear algebra, which is essential for modeling financial systems and evaluating risk. You will learn how to use matrices to represent financial data, and use linear transformations to model financial relationships. This course will help you develop the skills you need to succeed as a Quantitative Analyst.
See salaries and explore the career path for Quantitative Analyst
Machine Learning Engineer
A Machine Learning Engineer develops and deploys machine learning models to solve real-world problems. This course may be useful for a Machine Learning Engineer by strengthening your understanding of linear algebra, which is essential for building and training machine learning models. You will learn how to use matrices to represent data, and use linear transformations to model relationships between variables. This course will provide you with the mathematical foundation you need to succeed as a Machine Learning Engineer.
See salaries and explore the career path for Machine Learning Engineer
Statistician
A Statistician collects, analyzes, and interprets data to help businesses and organizations make informed decisions. This course can be useful for a Statistician by enhancing your understanding of linear algebra, which is essential for statistical modeling and analysis. You will learn how to use matrices to represent and manipulate data, and use linear transformations to model statistical relationships. This course will provide you with the skills you need to succeed as a Statistician.
See salaries and explore the career path for Statistician
Biostatistician
A Biostatistician applies mathematical and statistical techniques to solve problems in biology and medicine. This course can be useful for a Biostatistician by enhancing your understanding of linear algebra, which is essential for analyzing complex biological data. You will be able to use matrices to represent and manipulate data, and use linear transformations to model biological systems. This course will provide you with the mathematical foundation you need to succeed as a Biostatistician.
See salaries and explore the career path for Biostatistician
Data Scientist
A Data Scientist collects, analyzes, and interprets data to help businesses make informed decisions. This course can be useful for a Data Scientist by enhancing your understanding of linear algebra, which is essential for working with large datasets. You will learn how to use matrices to organize and manipulate data, and use linear transformations to model relationships between variables. This course will help you develop the analytical skills you need to succeed as a Data Scientist.
See salaries and explore the career path for Data Scientist
Actuary
An Actuary uses mathematical and statistical techniques to assess and manage risk in the insurance and finance industries. This course may be useful for an Actuary by enhancing your understanding of linear algebra, which is used in pricing insurance policies and evaluating financial risk. You will learn how to use matrices to represent and manipulate data, and use linear transformations to model financial systems. This course will help you develop the skills you need to succeed as an Actuary.
See salaries and explore the career path for Actuary
Software Engineer
A Software Engineer designs, develops, and maintains software applications. This course may be useful for a Software Engineer by enhancing your understanding of linear algebra, which is used in computer graphics, image processing, and artificial intelligence. You will learn how to use matrices to represent and manipulate data, and use linear transformations to model complex systems. This course will provide you with the skills you need to succeed as a Software Engineer.
See salaries and explore the career path for Software Engineer
Operations Research Analyst
An Operations Research Analyst uses mathematical and statistical techniques to solve problems in business and industry. This course can be useful for an Operations Research Analyst by enhancing your understanding of linear algebra, which is used in optimization and decision-making. You will learn how to use matrices to represent and manipulate data, and use linear transformations to model business systems. This course will provide you with the skills you need to succeed as an Operations Research Analyst.
See salaries and explore the career path for Operations Research Analyst
Financial Risk Manager
A Financial Risk Manager uses mathematical and statistical techniques to assess and manage risk in the financial industry. This course may be useful for a Financial Risk Manager by strengthening your understanding of linear algebra, which is used in modeling financial systems and evaluating risk. You will learn how to use matrices to represent financial data, and use linear transformations to model financial relationships. This course will help you develop the skills you need to succeed as a Financial Risk Manager.
See salaries and explore the career path for Financial Risk Manager
Investment Analyst
An Investment Analyst evaluates and manages investment portfolios for individuals and institutions. This course may be useful for an Investment Analyst by enhancing your understanding of linear algebra, which is used in financial modeling and risk assessment. You will learn how to use matrices to represent financial data, and use linear transformations to model financial systems. This course will help you develop the skills you need to succeed as an Investment Analyst.
See salaries and explore the career path for Investment Analyst
Data Analyst
A Data Analyst collects, analyzes, and interprets data to help businesses make informed decisions. This course may be useful for a Data Analyst by strengthening your skills in linear algebra, which is important for working with large datasets. You will learn how to use matrices to organize and manipulate data, and use linear transformations to model relationships between variables. This course will help you develop the analytical skills you need to succeed as a Data Analyst.
See salaries and explore the career path for Data Analyst
Market Researcher
A Market Researcher collects and analyzes data to help businesses understand their customers and make informed decisions. This course may be useful for a Market Researcher by strengthening your understanding of linear algebra, which is used in statistical modeling and analysis. You will learn how to use matrices to organize and manipulate data, and use linear transformations to model relationships between variables. This course will help you develop the analytical skills you need to succeed as a Market Researcher.
See salaries and explore the career path for Market Researcher
Operations Manager
An Operations Manager plans and executes the operations of a business or organization. This course may be useful for an Operations Manager by enhancing your understanding of linear algebra, which is used in optimization and decision-making. You will learn how to use matrices to represent and manipulate data, and use linear transformations to model business systems. This course will provide you with the skills you need to succeed as an Operations Manager.
See salaries and explore the career path for Operations Manager
Product Manager
A Product Manager plans and executes the development and launch of new products or services. This course may be useful for a Product Manager by enhancing your understanding of linear algebra, which is used in market research and customer segmentation. You will learn how to use matrices to organize and manipulate data, and use linear transformations to model customer relationships. This course will help you develop the analytical skills you need to succeed as a Product Manager.
See salaries and explore the career path for Product Manager
Financial Analyst
A Financial Analyst uses mathematical and statistical techniques to evaluate investments and make financial decisions. This course may be useful for a Financial Analyst by enhancing your understanding of linear algebra, which is used in financial modeling and risk assessment. You will learn how to use matrices to represent financial data, and use linear transformations to model financial systems. This course will help you develop the skills you need to succeed as a Financial Analyst.
See salaries and explore the career path for Financial Analyst

Reading list

We've selected 11 books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Linear Algebra: Linear Systems and Matrix Equations.
Cover image
Introduction to Algebraic Geometry
Save
Provides a rigorous and comprehensive treatment of linear algebra, suitable for students with a strong mathematical background. It covers topics such as vector spaces, linear transformations, and abstract algebra.
Introduction to Algebraic Geometry
Paperback
Check
Introduction to Algebraic Geometry
Kindle Edition
Check
Cover image
An Introduction to the Language of Category Theory
Save
Provides an advanced treatment of linear algebra, suitable for students with a strong mathematical background. It covers topics such as vector spaces, linear transformations, and abstract algebra.
An Introduction to the Language of Category Theory
Paperback
Check
An Introduction to the Language of Category Theory
Kindle Edition
Check
Cover image
Matrix Mathematics
Save
Provides a comprehensive treatment of matrix analysis, suitable for students with a strong mathematical background. It covers topics such as matrix norms, eigenvalues, and singular value decomposition.
Matrix Mathematics: A Second Course in Linear...
Hardcover
$$$
Matrix Mathematics: A Second Course in Linear...
Kindle Edition
$$$
Cover image
Linear Algebra and Geometry
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Provides a comprehensive treatment of linear algebra and geometry, suitable for students with a strong mathematical background. It covers topics such as vector spaces, linear transformations, and abstract algebra.
Linear Algebra and Geometry
Paperback
Check
Linear Algebra and Geometry
Kindle Edition
Check
Cover image
Linear Algebra Done Right
Save
Provides a more advanced treatment of linear algebra, suitable for students with a strong mathematical background. It covers topics such as vector spaces, linear transformations, and abstract algebra.
Linear Algebra Done Right (Undergraduate Texts in...
Hardcover
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Linear Algebra Done Right (Undergraduate Texts in...
Kindle Edition
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Linear Algebra Done Right (Undergraduate Texts in...
Hardcover
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Linear Algebra Done Right (Undergraduate Texts in...
Paperback
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(Español) Aljebra lineala ondo egina
Paperback
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Cover image
Modern Analysis (1997)
Save
Provides a comprehensive treatment of linear algebra, suitable for students with a strong mathematical background. It covers topics such as vector spaces, linear transformations, and abstract algebra.
Modern Analysis (1997) (CRC Press Revivals)
Kindle Edition
$$$
Linear Algebra and Its Applications
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Provides a comprehensive introduction to linear algebra, covering a wide range of topics including matrices, vectors, linear transformations, and eigenvalues. It is written in a clear and concise style, with plenty of examples and exercises to help students understand the concepts.
Linear Algebra and Its Applications
Paperback
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Linear Algebra and Its Applications
Kindle Edition
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Introduction to Linear Algebra
Save
Provides a clear and concise introduction to linear algebra. It covers the basics of linear algebra, including matrices, vectors, linear transformations, and eigenvalues.
Introduction to Linear Algebra (Gilbert Strang)
Hardcover
$$$$
Cover image
Linear Algebra
Save
Provides a geometric approach to linear algebra, suitable for students with a strong mathematical background. It covers topics such as vector spaces, linear transformations, and eigenvalues.
Linear Algebra: A Geometric Approach
Hardcover
$$$$
Linear Algebra: A Geometric Approach
Kindle Edition
$$$$
Linear Algebra with Applications
Save
Provides a practical introduction to linear algebra, with a focus on applications in engineering, science, and economics. It covers topics such as matrices, vectors, linear transformations, and eigenvalues.
Linear Algebra with Applications
Paperback
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Linear Algebra with Applications
Kindle Edition
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Cover image
Linear Algebra for Everyone
Save
Provides a practical introduction to linear algebra, with a focus on applications in engineering, science, and economics. It covers topics such as matrices, vectors, linear transformations, and eigenvalues.
Linear Algebra for Everyone
Paperback
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Linear Algebra for Everyone
Kindle Edition
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Course image
Level
Introductory
Via
Coursera
Institution
Johns Hopkins University
Instructor
Joseph W. Cutrone, PhD
Language
English
This course belongs to a collection called Linear Algebra from Elementary to Advanced. It's comprised of three courses:
  1. Linear Algebra: Linear Systems and Matrix Equations (viewing)
  2. Linear Algebra: Orthogonality and Diagonalization
  3. Linear Algebra: Matrix Algebra, Determinants, & Eigenvectors
Traffic lights
Read about what's good
what should give you pause
and possible dealbreakers
Develops fluency in the language of linear algebra, which is highly relevant in math, engineering, and the sciences
Introduces the fundamental concepts of linear equations, matrix methods, analytical geometry, and linear transformations
Provides a strong foundation for those interested in pursuing advanced study in linear transformations
Emphasizes both theory and applications, providing learners with valuable skills and understanding
Taught by Joseph W. Cutrone, PhD, an expert in the field of linear algebra
Requires a prior understanding of algebra
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