Introduction to Linear Algebra from Coursera

Linear algebra and calculus are the two most important foundational pillars on which modern mathematics is built. They are studied by almost all mathematics students at university, though typically labelled as different subjects and taught in parallel. Over time, students discover that linear algebra and calculus are inseparable (but not identical) twins that interlock to form the backbone of almost all applications of mathematics to physical and biological sciences, engineering and computer science. It is recommended that participants in the MOOC Introduction to Linear Algebra have already taken, or take in parallel, the MOOC Introduction to Calculus.

All of our modern technical and electronic systems, such as the internet and search engines, on which we rely and tend to take for granted in our daily lives, work because of methods and techniques adapted from classical linear algebra. The key ideas involve vector and matrix arithmetic as well as clever methods for working around or overcoming difficulties, a form of obstacle avoidance, articulated in this course as the Conjugation Principle.

This course emphasises geometric intuition, gradually introducing abstraction and algebraic and symbolic manipulation, while at the same time striking a balance between theory and application, leading to a mastery of key threshold concepts in foundational mathematics.

Students taking Introduction to Linear Algebra will:

• gain familiarity with the arithmetic of geometric vectors, which may be thought of as directed line segments that can move about freely in space, and can be combined in different ways, using vector addition, scalar multiplication and two types of multiplication, the dot and cross product, related to projections and orthogonality (first week),

• develop fluency with lines and planes in space, represented by vector and Cartesian equations, and learn how to solve systems of equations, using the method of Gaussian elimination and introduction of parameters, using fields of real numbers and modular arithmetic with respect to a prime number (second week),

• be introduced to and gain familiarity with matrix arithmetic, matrix inverses, the role of elementary matrices and their relationships with matrix inversion and systems of equations, calculations and theory involving determinants (third week),

• be introduced to the theory of eigenvalues and eigenvectors, how they are found or approximated, and their role in diagonalisation of matrices (fourth week),

• see applications to simple Markov processes and stochastic matrices, and an introduction to linear transformations, illustrated using dilation, rotation and reflection matrices (fourth week),

• see a brief introduction to the arithmetic of complex numbers and discussion of the Fundamental Theorem of Algebra (fourth week).

What's inside

Syllabus

Week 1 - Geometric Vectors in the Plane and in Space

This module introduces and explores the useful and elegant arithmetic of geometric vectors, regarded as directed line segments, which move about freely in the plane and in space. Objects in this arithmetic are ubiquitous throughout the physical world, modelling vector quantities. Students acquire tools that enable them to explore precise geometrical relationships between objects, prove difficult theorems and solve optimisation problems. This arithmetic forms a prototype for the general and abstract theory of vector spaces, developing students’ intuition and preparing them for advanced courses on linear algebra.

Week 2 - Lines and Planes in Space and Systems of Linear Equations

This module introduces and develops fluency with lines and planes in space, represented by vector and Cartesian equations. Students learn how to exploit the arithmetic of geometric vectors to solve difficult optimisation problems such as finding the closest point to a plane or finding the closest points on a pair of skew lines. The module introduces systems of linear equations and then develops the method of Gaussian elimination, using elementary row operations, followed by back substitution, to express solutions in terms of parameters. Systems of equations are also explored and solved using modular arithmetic with respect to a given prime number.

Week 3 - Matrix Arithmetic and the Theory of Determinants

This module introduces matrix arithmetic and the theory of determinants. Students first learn how to add matrices of the same size and how to multiply by a scalar. They then learn how to multiply matrices of compatible sizes, using cascades of dot products of rows with columns. This is described succinctly using Sigma notation, which is then used to prove associativity of matrix multiplication. Students learn about elementary matrices, which are basic building blocks in matrix arithmetic, closely related to elementary row operations used in Gaussian elimination. Students learn about matrix inverses and how to find them. Students learn about determinants and their properties, including the multiplicative property and a simple criterion for recognising invertibility of a matrix. Students see applications to cross products of vectors and for exploring spatial relationships between points and triangles.

Week 4 - Eigentheory and Diagonalisation

This module continues the development of matrix arithmetic by introducing eigenvalues and associated eigenvectors. Interpreted geometrically, these allow one to find directions in which a given linear operator associated with a matrix moves vectors in straight lines. This leads to the technique of diagonalisation, enabling one to solve difficult problems in matrix arithmetic, including finding formulae for powers of a given square matrix. This has many applications, including, for example, in exploring the behaviour of a Markov process described by a stochastic matrix, the mathematics of which underlies search engines on the internet. Eigenvalues can be found, in principle, by solving the characteristic equation of a matrix. Associated eigenvectors can then be found by solving an associated homogeneous system of equations. In practice, there are iterative numerical techniques for finding approximations of eigenvalues and eigenvectors, using a technique associated with Perron’s Theorem. Diagonalisation is a manifestation of the general Conjugation Principle, explored in different contexts. Linear transformations are introduced, focusing on transformation of the plane. Rotations and reflections of the plane combine to form the two-dimensional orthogonal group. Scalar dilations and rotations combine to form a copy of the field of complex numbers. A sketch of Smale’s proof of the Fundamental Theorem of Algebra is given, which says that any nonconstant polynomial with complex coefficients has a complex root, so that all square matrices have eigenvalues, when working over the field of complex numbers.

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Examines the arithmetic of linear vectors and develops fluency with lines and planes in space, which is a strong foundation for students

Develops fluency with matrix arithmetic, matrix inverses, and the theory of determinants, which are essential concepts in linear algebra

Introduces the theory of eigenvalues and eigenvectors, which are crucial for understanding linear transformations and diagonalization of matrices

Provides applications to simple Markov processes and stochastic matrices, as well as an introduction to linear transformations, which are widely used in various fields

Emphasizes geometric intuition and gradually introduces abstraction, providing a clear learning path for students

Assumes participants in the MOOC Introduction to Linear Algebra have already taken, or take in parallel, the MOOC Introduction to Calculus, which might not be accessible to all

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 Introduction to Linear Algebra with these activities:

Review matrix algebra

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Reviewing the basics of matrix algebra will provide a stronger foundation for understanding the concepts covered in this course.

Browse courses on Matrix Arithmetic

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Review the basic operations of matrix arithmetic, including addition, subtraction, and multiplication.
Practice solving systems of linear equations using matrix methods.
Review the concepts of matrix inverses and determinants.

Watch video tutorials on linear algebra

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Watching video tutorials can provide a different perspective on the concepts covered in the course and help you reinforce your understanding.

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Find video tutorials on linear algebra that cover the topics you are studying in the course.
Watch the tutorials carefully and take notes on the key concepts.
Pause the tutorials and try to solve the problems yourself before watching the solution.

Solve systems of linear equations

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Practicing solving systems of linear equations will help you develop fluency and confidence in this fundamental skill.

Browse courses on Systems of Linear Equations

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Find the solutions to systems of linear equations using Gaussian elimination.
Use Cramer's rule to solve systems of linear equations.
Apply systems of linear equations to solve real-world problems.

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Create a cheat sheet for matrix algebra

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Creating a cheat sheet will help you solidify your understanding of the key concepts and formulas in matrix algebra.

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Summarize the key concepts and formulas for matrix algebra.
Organize the information in a clear and concise way.
Review your cheat sheet regularly to reinforce your memory.

Join a study group for linear algebra

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Joining a study group can provide you with opportunities to discuss the material with other students and get help with concepts you are struggling with.

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Find a study group that meets regularly to discuss linear algebra.
Attend the study group meetings regularly and participate in the discussions.
Ask questions and share your insights with the other members of the group.

Read 'Linear Algebra Done Right' by Sheldon Axler

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This book provides a comprehensive and rigorous treatment of linear algebra, which can help you deepen your understanding of the subject.

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

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Read the book carefully and take notes on the key concepts.
Work through the exercises at the end of each chapter to test your understanding.
Use the book as a reference when you need to review specific topics.

Write a paper on a topic in linear algebra

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Writing a paper on a topic in linear algebra will challenge you to think critically about the material and demonstrate your understanding.

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Choose a topic in linear algebra that you are interested in.
Research the topic thoroughly and gather information from multiple sources.
Write a paper that presents your research in a clear and concise way.

Contribute to an open-source linear algebra library

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Contributing to an open-source linear algebra library can help you gain practical experience and deepen your understanding of the subject.

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Find an open-source linear algebra library that you are interested in.
Read the documentation and learn how to use the library.
Contribute to the library by adding new features or fixing bugs.

Career center

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

Mathematician

Mathematicians develop and apply mathematical theories and techniques to solve problems in a variety of fields. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding mathematical theories and techniques. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Mathematicians, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Mathematician

Machine Learning Engineer

Machine Learning Engineers develop and implement machine learning algorithms. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding the mathematical foundations of machine learning. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Machine Learning Engineers, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Machine Learning Engineer

Operations Research Analyst

Operations Research Analysts use mathematical models to improve the efficiency of systems and processes. This course provides a strong foundation in linear algebra and calculus, which are essential for developing and analyzing these models. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Operations Research Analysts, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Operations Research Analyst

Statistician

Statisticians use mathematical and statistical techniques to collect, analyze, and interpret data. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding statistical models and making sound decisions based on data. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Statisticians, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Statistician

Data Analyst

Data Analysts use mathematical and statistical techniques to analyze data and extract insights. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding data analysis techniques and making sound decisions based on data. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Data Analysts, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Data Analyst

Data Scientist

Data Scientists use mathematical and statistical techniques to analyze data and extract insights. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding data analysis techniques and making sound decisions based on data. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Data Scientists, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Data Scientist

Computer Scientist

Computer Scientists design, develop, and maintain software applications and systems. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding computer science principles and models. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Computer Scientists, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Computer Scientist

Physicist

Physicists use mathematical and computational techniques to study the fundamental laws of nature. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding physical theories and models. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Physicists, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Physicist

Engineer

Engineers design, develop, and maintain systems and structures. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding engineering principles and models. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Engineers, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Engineer

Computational Scientist

Computational Scientists apply their knowledge of linear algebra and calculus to develop and analyze mathematical models that simulate complex systems. This course provides a strong foundation in these mathematical concepts, which are essential for success in this field. The course's emphasis on geometric intuition and problem-solving skills will also be beneficial to Computational Scientists, who often need to visualize and solve complex problems.

See salaries and explore the career path for Computational Scientist

Software Engineer

Software Engineers design, develop, and maintain software applications. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding the mathematical foundations of computer science. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Software Engineers, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Software Engineer

Economist

Economists use mathematical and statistical techniques to analyze economic data and make economic forecasts. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding economic models and making sound economic forecasts. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Economists, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Economist

Financial Analyst

Financial Analysts use mathematical and statistical techniques to analyze financial data and make investment recommendations. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding financial models and making sound investment decisions. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Financial Analysts, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Financial Analyst

Quantitative Analyst

Quantitative Analysts use mathematical and statistical techniques to analyze financial data and make investment decisions. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding financial models and making sound investment decisions. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Quantitative Analysts, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Quantitative Analyst

Actuary

Actuaries use mathematical and statistical techniques to assess risk and make financial decisions. This course provides a strong foundation in linear algebra and calculus, which are essential for understanding actuarial models and making sound financial decisions. The course's emphasis on problem-solving skills and geometric intuition will also be beneficial to Actuaries, who often need to solve complex problems in a variety of settings.

See salaries and explore the career path for Actuary

Introduction to Linear Algebra

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