Linear Algebra I: Linear Equations from edX

Systems of equations live at the heart of linear algebra. In this course you will explore fundamental concepts by exploring definitions and theorems that give a basis for this subject. At the start of this course we introduce systems of linear equations and a systematic method for solving them. This algorithm will be used for computations throughout the course as you investigate applications of linear algebra and more complex algorithms for analyzing them.

Later in this course you will later see how a system of linear equations can be represented in other ways, which can reduce problems involving linear combinations of vectors to approaches that involve systems of linear equations. Towards the end of the course we explore linear independence and linear transformations. They have an essential role throughout our course and in applications of linear algebra to many areas of industry, science, and engineering. __

What you'll learn

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

Evaluate mathematical expressions to compute quantities that deal with linear systems
Characterize a linear system in terms of the number of solutions, and whether the system is consistent or inconsistent.
Apply elementary row operations to solve linear systems of equations.
Characterize a set of vectors in terms of linear combinations, their span, and how they are related to each other geometrically
Characterize a set of vectors and linear systems using the concept of linear independence.
Construct dependence relations between linearly dependent vectors.
Identify and construct linear transformations of a matrix.
Characterize linear transformations as onto and/or one-to-one.

What's inside

Learning objectives

Evaluate mathematical expressions to compute quantities that deal with linear systems
Characterize a linear system in terms of the number of solutions, and whether the system is consistent or inconsistent.
Apply elementary row operations to solve linear systems of equations.
Characterize a set of vectors in terms of linear combinations, their span, and how they are related to each other geometrically

Characterize a set of vectors and linear systems using the concept of linear independence.
Construct dependence relations between linearly dependent vectors.
Identify and construct linear transformations of a matrix.
Characterize linear transformations as onto and/or one-to-one.

Evaluate mathematical expressions to compute quantities that deal with linear systems
Characterize a linear system in terms of the number of solutions, and whether the system is consistent or inconsistent.
Apply elementary row operations to solve linear systems of equations.
Characterize a set of vectors in terms of linear combinations, their span, and how they are related to each other geometrically
Characterize a set of vectors and linear systems using the concept of linear independence.
Construct dependence relations between linearly dependent vectors.
Identify and construct linear transformations of a matrix.
Characterize linear transformations as onto and/or one-to-one.

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Develops the foundation for a strong understanding of linear algebra concepts

Teaches fundamental concepts of linear algebra, providing a solid foundation in the subject

Emphasizes problem-solving techniques and computational methods, preparing learners for real-world applications of linear algebra

Introduces students to the basics of systems of linear equations, leading to a comprehensive understanding of the topic

Focuses on developing learners' ability to analyze and describe systems of linear equations, applicable in various fields

Taught by Greg Mayer, an experienced instructor with a strong track record in linear algebra education

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 I: Linear Equations with these activities:

Review of Matrix Operations and Properties

Show steps

Reinforce your understanding of matrix operations and properties to lay a solid foundation for linear algebra.

Browse courses on Matrix Operations

Show steps

Go through your notes or textbooks to review the concepts of matrix operations (addition, subtraction, multiplication, transpose) and properties (determinants, inverses).
Solve practice problems to apply and strengthen your understanding of these operations and properties.

Tutorial on Solving Linear Systems Ax=b

Show steps

Reinforce your understanding of solving linear systems Ax=b by following step-by-step tutorials.

Show steps

Search for tutorials on solving linear systems Ax=b (e.g., Khan Academy, Coursera, Udacity).
Watch the tutorials carefully, taking notes and making sure to understand each step.
Practice solving linear systems Ax=b using the methods explained in the tutorials.

Collaboration on solving linear equations

Show steps

Engage with peers to discuss and solve linear equations, enhancing your understanding through collaboration.

Browse courses on Linear Equations

Show steps

Form study groups with peers.
Select practice problems or linear equations to solve.
Discuss and solve the problems together as a group.
Share different approaches and insights.

Five other activities

Expand to see all activities and additional details

Show all eight activities

Solve linear systems using row operations

Show steps

Familiarize yourself with elementary row operations to solve linear systems effectively.

Browse courses on Linear Systems

Show steps

Review the steps for performing elementary row operations (e.g., swapping rows, multiplying rows by constants, adding multiples of one row to another).
Practice applying elementary row operations to simplify linear systems.
Solve linear systems using row operations.

Project: Create a linear algebra cheat sheet

Show steps

Challenge yourself by creating a cheat sheet that summarizes key concepts and formulas in linear algebra, reinforcing your understanding.

Browse courses on Linear Algebra

Show steps

Review the course materials to identify essential concepts and formulas in linear algebra.
Organize the concepts and formulas logically.
Create the cheat sheet using a visually appealing format.
Use the cheat sheet as a quick reference guide to refresh your memory.

Workshop on Linear Algebra Fundamentals

Show steps

Attend a workshop to deepen your understanding of linear algebra, exploring advanced concepts beyond the course.

Browse courses on Linear Algebra

Show steps

Research and identify relevant linear algebra workshops.
Register for the workshop.
Actively participate in the workshop, taking notes and asking questions.
Apply the knowledge and techniques learned to solve problems and enhance your comprehension.

Mentor a student in introductory linear algebra

Show steps

Solidify your understanding of linear algebra by mentoring a student, providing support, and clarifying concepts.

Browse courses on Linear Algebra

Show steps

Offer your services as a mentor through platforms or connect with students seeking help.
Communicate with the student to assess their needs and goals.
Prepare and deliver tailored explanations and guidance on linear algebra concepts.
Provide feedback and encourage the student's problem-solving skills.

Project: Develop a mathematical model using linear equations

Show steps

Apply your knowledge of linear equations to a real-world scenario, fostering critical thinking and problem-solving skills.

Browse courses on Linear Equations

Show steps

Identify a problem or situation that can be mathematically modeled using linear equations.
Develop the mathematical model by setting up a system of linear equations.
Solve the system of linear equations using appropriate methods (e.g., Gaussian elimination, Cramer's rule).
Analyze the solution and interpret the results in the context of the original problem.

Career center

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

Quantitative Analyst

Quantitative Analysts use mathematical models to analyze financial data. They build financial models that help predict trends in the financial market. This course can help someone who wants to become a Quantitative Analyst because it provides a solid foundation in linear algebra, a branch of mathematics that is essential for building financial models.

See salaries and explore the career path for Quantitative Analyst

Operations Research Analyst

Operations Research Analysts use mathematical models to solve problems in business and industry. They may work on projects such as improving the efficiency of a manufacturing process or optimizing the routing of delivery trucks. This course may be useful for someone who wants to become an Operations Research Analyst because it provides a foundation in linear algebra, a branch of mathematics that is essential for building mathematical models.

See salaries and explore the career path for Operations Research Analyst

Data Scientist

Data Scientists use mathematical and statistical techniques to analyze data and extract insights from it. They may work on projects such as developing predictive models or identifying trends in customer behavior. This course may be useful for someone who wants to become a Data Scientist because it provides a foundation in linear algebra, a branch of mathematics that is essential for data analysis.

See salaries and explore the career path for Data Scientist

Software Engineer

Software Engineers design, develop, and maintain software applications. They may work on projects such as developing new features for a website or creating a mobile app. This course may be useful for someone who wants to become a Software Engineer because it provides a foundation in linear algebra, a branch of mathematics that is used in computer graphics and other areas of software development.

See salaries and explore the career path for Software Engineer

Financial Analyst

Financial Analysts use financial data to make investment recommendations. They may work for investment banks, hedge funds, or other financial institutions. This course may be useful for someone who wants to become a Financial Analyst because it provides a foundation in linear algebra, a branch of mathematics that is used in financial modeling.

See salaries and explore the career path for Financial Analyst

Actuary

Actuaries use mathematical and statistical techniques to assess risk and uncertainty. They may work for insurance companies, pension funds, or other financial institutions. This course may be useful for someone who wants to become an Actuary because it provides a foundation in linear algebra, a branch of mathematics that is used in risk assessment.

See salaries and explore the career path for Actuary

Economist

Economists use economic data to analyze economic trends and make predictions about the future. They may work for government agencies, think tanks, or other organizations. This course may be useful for someone who wants to become an Economist because it provides a foundation in linear algebra, a branch of mathematics that is used in economic modeling.

See salaries and explore the career path for Economist

Statistician

Statisticians use statistical techniques to analyze data and extract insights from it. They may work for government agencies, businesses, or other organizations. This course may be useful for someone who wants to become a Statistician because it provides a foundation in linear algebra, a branch of mathematics that is used in statistical modeling.

See salaries and explore the career path for Statistician

Market Researcher

Market Researchers use research methods to collect and analyze data about consumer behavior. They may work for marketing firms, advertising agencies, or other organizations. This course may be useful for someone who wants to become a Market Researcher because it provides a foundation in linear algebra, a branch of mathematics that is used in data analysis.

See salaries and explore the career path for Market Researcher

Business Analyst

Business Analysts use data to analyze business processes and make recommendations for improvement. They may work for consulting firms, corporations, or other organizations. This course may be useful for someone who wants to become a Business Analyst because it provides a foundation in linear algebra, a branch of mathematics that is used in data analysis.

See salaries and explore the career path for Business Analyst

Consultant

Consultants provide advice and guidance to businesses and organizations. They may work on projects such as developing new strategies, improving operations, or implementing new technologies. This course may be useful for someone who wants to become a Consultant because it provides a foundation in linear algebra, a branch of mathematics that is used in data analysis and problem solving.

See salaries and explore the career path for Consultant

Teacher

Teachers educate students at all levels, from elementary school to college. They may teach a variety of subjects, including math, science, and social studies. This course may be useful for someone who wants to become a Teacher because it provides a foundation in linear algebra, a branch of mathematics that is taught in high school and college.

See salaries and explore the career path for Teacher

Insurance Underwriter

Insurance Underwriters assess risk and determine insurance rates. They may work for insurance companies or insurance brokerages. This course may be useful for someone who wants to become an Insurance Underwriter because it provides a foundation in linear algebra, a branch of mathematics that is used in risk assessment.

See salaries and explore the career path for Insurance Underwriter

Risk Manager

Risk Managers identify, assess, and manage risks for businesses and organizations. They may work for corporations, insurance companies, or other organizations. This course may be useful for someone who wants to become a Risk Manager because it provides a foundation in linear algebra, a branch of mathematics that is used in risk assessment.

See salaries and explore the career path for Risk Manager

Auditor

Auditors examine financial records to ensure accuracy and compliance with laws and regulations. They may work for accounting firms, government agencies, or other organizations. This course may be useful for someone who wants to become an Auditor because it provides a foundation in linear algebra, a branch of mathematics that is used in financial analysis.

See salaries and explore the career path for Auditor