Linear Algebra I: Vectors and Linear Equations 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 vectors (from both algebraic and geometric perspectives) and solving linear equations. It 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 or 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.

This course is part of our series Mastering Mathematics for Engineers , and together with the course Linear Algebra II part of the program Mastering Linear Algebra.

What's inside

Learning objectives

Apply the dot product and cross product.
Describe lines, planes and their intersections.
Solve systems of linear equations and describe the solution set.
Decide whether vectors are linear dependent or not.

Recognize linear subspaces, describe elements of linear subspaces using bases and coordinates.
Calculate projections and orthogonal decompositions of vectors.
Find least-square solutions of a system of linear equations and apply it to regression.

Apply the dot product and cross product.
Describe lines, planes and their intersections.
Solve systems of linear equations and describe the solution set.
Decide whether vectors are linear dependent or not.
Recognize linear subspaces, describe elements of linear subspaces using bases and coordinates.
Calculate projections and orthogonal decompositions of vectors.
Find least-square solutions of a system of linear equations and apply it to regression.

Syllabus

Week 1: Vectors

calculating with vectors

the dot product

the cross product

lines and planes

Week 2: Linear equations

systems of equations

solving systems of equations

structure of the solutions set

Week 3: Linear dependence

linear combinations

linear dependence

relations between concepts

Week 4: Linear subspaces

What are linear subspaces?

basis and coordinates

dimension

the rank theorem

Week 5: Orthogonality

orthogonal sets

orthogonal projections

the Gram-Schmidt algorithm

orthogonal complements

transposition

Week 6: Least square solutions

"solving” an inconsistent system

normal equations

application to regression

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Appropriate for students who need a refresher on the fundamentals of linear algebra

Focuses on vectors, linear equations, and linear subspaces, which are essential concepts in linear algebra

Provides plenty of exercises and immediate feedback through the Grasple platform

May be more suitable for students who are already familiar with the basics 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 I: Vectors and Linear Equations with these activities:

Watch Video Lectures on Linear Algebra

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Expand your understanding of linear algebra concepts through engaging video lectures.

Browse courses on Linear Algebra

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Identify reputable sources for linear algebra video lectures
Watch and take notes on the lectures covering key concepts
Review the material and clarify any doubts

Read and Review Linear Algebra Done Right

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Review the fundamentals of linear algebra with this comprehensive and accessible text.

View Linear Algebra Done Right (Undergraduate Texts... on Amazon

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Read and understand the key concepts introduced in each chapter
Solve the end-of-chapter exercises to test your understanding

Connect with an Experienced Linear Algebra Professional

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Gain valuable insights and guidance from someone who has expertise in linear algebra.

Browse courses on Linear Algebra

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Identify potential mentors who work in fields that utilize linear algebra
Reach out to them and express your interest in connecting
Set up a meeting to discuss your goals and seek their advice

Five other activities

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Create a Mind Map of Linear Algebra Concepts

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Visualize and organize your understanding of linear algebra by creating a mind map.

Browse courses on Linear Algebra

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Identify the main branches and topics of linear algebra
Create a central node for each main topic
Connect related subtopics to the corresponding main topic nodes

Solve Linear Algebra Practice Problems

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Sharpen your problem-solving skills by tackling a variety of linear algebra problems.

Browse courses on Linear Algebra

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Identify the key concepts involved in the problem
Apply the appropriate mathematical techniques to solve for the unknown variables
Check your solution for accuracy

Join a Study Group for Linear Algebra

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Collaborate with peers to reinforce your understanding and tackle challenging concepts.

Browse courses on Linear Algebra

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Find a group of students taking the same linear algebra course
Meet regularly to discuss course material
Work together to solve problems and prepare for exams

Develop a Linear Algebra Library in Your Preferred Programming Language

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Solidify your understanding of linear algebra by implementing its core concepts in code.

Browse courses on Linear Algebra

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Familiarize yourself with a programming language that supports linear algebra operations
Design and implement functions for performing matrix operations, vector operations, and solving linear systems
Test your library thoroughly and debug any errors

Write a Detailed Summary of a Linear Algebra Research Paper

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Deepen your understanding of advanced linear algebra concepts by analyzing and summarizing research.

Browse courses on Linear Algebra

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Identify a research paper in a specialized area of linear algebra
Read and understand the paper thoroughly
Extract the key findings, methods, and implications of the research
Write a detailed summary that clearly articulates the significance of the paper

Career center

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

Actuary

Actuaries use mathematics and statistics to assess risk and uncertainty, especially in the insurance and finance industries. They use linear algebra to model complex systems and predict future events. By enrolling in this course, you'll gain valuable knowledge in the mathematics behind actuarial science and take steps towards a successful career in this field.

See salaries and explore the career path for Actuary

Operations Research Analyst

Operations Research Analysts use mathematical and analytical methods to solve complex business problems. They use linear algebra and other advanced mathematical principles to model and analyze systems in order to identify potential bottlenecks and inefficiencies. Take this course to boost your problem solving skills and gain the knowledge needed to succeed as an Operations Research Analyst.

See salaries and explore the career path for Operations Research Analyst

Machine Learning Engineer

Machine Learning Engineers are responsible for designing, developing, and deploying machine learning models. They use their knowledge of linear algebra to build models that can learn from data and make predictions. This course will help you build the skills needed to start your journey as a Machine Learning Engineer. Gain experience with vectors and linear combinations to prepare today!

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Data Analyst

Begin your career as a Data Analyst by taking this course! The primary responsibility of a Data Analyst is to analyze data and find helpful insights to guide business decisions. They use advanced tools to clean, manage and analyze large volumes of data, which is a skill that can be learned through this course. Enrolling in this course reveals that you are dedicated to expanding your skillset and are eager to contribute your talents as a Data Analyst.

See salaries and explore the career path for Data Analyst

Data Engineer

Data Engineers are responsible for building and maintaining data pipelines that collect, store, and transform data for analysis. They use linear algebra to design and implement these pipelines and to ensure that the data is accurate and reliable. Begin your journey to a rewarding career by enrolling in this course and growing your proficiency in linear algebra and its applications in data engineering!

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Statistician

Statisticians collect, analyze, and interpret data to help businesses and organizations make informed decisions. They use statistical methods to uncover patterns and trends in data that would not be easily found otherwise. They also collaborate with other professionals to develop solutions to complex problems. This course will help you build a solid foundation in mathematics as it relates to data analysis, and is a great starting point to begin a career as a Statistician.

See salaries and explore the career path for Statistician

Systems Engineer

Systems Engineers integrate complex systems together to meet specific client or business needs. They work on projects where they must manage how different systems work together. Systems Engineers with knowledge of linear algebra can better understand how these systems communicate and interact with each other. The insights you will gain from this course will support your work in designing, building, and maintaining complex systems.

See salaries and explore the career path for Systems Engineer

Quantitative Analyst

Quantitative Analysts, also known as Quants, use mathematical and statistical methods to analyze financial data. Some of their responsibilities include developing models that can predict financial risks and pricing. Learn the linear algebra and statistics needed to become a Quant by taking this course. Enrolling in this course shows a dedication to the field and can help kick off a successful career.

See salaries and explore the career path for Quantitative Analyst

Software Architect

Software Architects design and develop the overall structure of software systems. They use linear algebra to model the system's architecture and to ensure that it is efficient and scalable. Step into the role of a Software Architect with this course that can help you build your understanding of the mathematical principles used in software design.

See salaries and explore the career path for Software Architect

Researcher

Researchers that focus on Computer Science or Mathematics are likely to use linear algebra as a foundation in their work. They develop and analyze algorithms, models, and theories in order to solve complex problems. From artificial intelligence to computer graphics, a background in linear algebra is a valuable asset to have for Researchers. Take this course to get started and grow your knowledge to succeed in a research role.

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Data Scientist

Data Scientists are similar to Data Analysts, however they are more focused on creating effective algorithms and predictive models. These models help businesses understand complex data and make intelligent decisions. This course will help you think like a Data Scientist, expanding your understanding of linear equations and other elements of data analysis. Gain knowledge on how to analyze data and get started in this field!

See salaries and explore the career path for Data Scientist

Financial Analyst

Financial Analysts research and interpret financial data to help businesses and individuals make informed investment decisions. Linear algebra may be used to measure risk and return, or to analyze the creditworthiness of a company. By taking this course, you will build a strong foundation in the mathematical principles used in finance. It can help you prepare for a career as a Financial Analyst.

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Teacher

Teachers who specialize in mathematics can make a meaningful impact in students' lives by providing an engaging and supportive learning environment. Students who take this course can bolster their knowledge of linear algebra, which is an important part of preparing them for higher-level mathematics studies and careers in STEM fields. Enrolling demonstrates your dedication to teaching math and will provide you with helpful teaching tools.

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Software Developer

A Software Developer is a professional who designs, develops, and maintains software systems. They use computer programming languages to write code that meets the needs of their clients. Linear algebra may be used to create 3D graphics for video games or computer simulations. By taking this course you will improve your understanding of the math that underpins software development and prepare for a role in this field.

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Web Developer

Web Developers are responsible for designing, developing, and maintaining websites. They use programming languages to create the look and functionality of a website. This course may be useful for Web Developers who work on 3D graphics for websites or mobile applications. By learning more about linear algebra, you can better understand how to design and implement these features on a website.

See salaries and explore the career path for Web Developer