Linear Algebra: Linear Systems and Matrix Equations from Coursera

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. .

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.

Vector and Matrix Equations

In this section we temporarily leave our discussion of linear systems to discuss vectors. These nx1 matrices are used in many contexts in physics, computer science and data science. We show in this section that answering questions about linear combinations turns out to be equivalent to solving a system of linear equations, underlying the deep connections of linear algebra. We then introduce the notion of a matrix as a function on vectors. Questions now about properties of the matrix as a function also turn out to be answered by solving a linear system. These connections between matrices as functions, vectors, and linear systems are sometimes why linear algebra is called the "theory of everything".

Linear Transformations

In this module, we study sets of vectors and functions on them. Understanding vectors and how to manipulate them via functions is quite useful in many areas, in particular, physics, computer science, math, and data science. The concept of linear dependence and linear independence is introduced along with the concept of a linear transformation. We will see when a linear transformation T can be represented by a matrix, how to find the matrix, and start to analyze the matrix to extract information about T. Pay careful attention to the new definitions in this section as they will be foundational to future modules!

Final Assessment

In this cumulative assessment, we will ask about the definitions, theorems, and examples shown so far. This is an opportunity to assess your knowledge of the content. The foundational material in this course about linear systems, matrices, and vectors, is key to understanding the more advanced theory and applications of linear algebra to follow. Do the best you can on the assessment and review any questions that are incorrect and learn from them. Good luck!

Good to know

Know what's good

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, 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

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

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Reinforce your foundational knowledge in precalculus to strengthen your understanding of linear algebra concepts.

Browse courses on Precalculus

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Review the unit circle and trigonometric identities.
Practice solving systems of equations.
Familiarize yourself with matrices and their operations.

Solve Linear Equation Systems Practice

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Improve your problem-solving skills and reinforce your understanding of solving linear equations by engaging in dedicated practice.

Browse courses on Linear Equations

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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

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Supplement your learning by exploring online video tutorials that provide clear explanations and visual demonstrations of linear algebra concepts.

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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

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Join a Study Group for Linear Algebra

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Collaborate with peers to enhance your understanding of linear algebra through discussions, problem-solving, and knowledge sharing.

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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

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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

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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

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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.

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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

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

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.

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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.

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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.

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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

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

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.

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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