Linear Algebra: Matrix Algebra, Determinants, & Eigenvectors from Coursera

What's inside

Syllabus

Matrix Algebra

In this module, we now look at what arithmetic operation we can perform on nxm matrices and how these operations correspond to operations on functions. In particular, we will view at matrix multiplication AB as a composition of function A(B(x)). In this way, algebraic properties like non-commutativity will become more apparent. We will also look for those matrices that are invertible. Since we no longer have the Horizontal Line Test, new tests for invertibility will be needed. This will lead to the study of the very important matrix invariant, the determinant.

Subspaces

In this module we investigate the structure of R^n by formally defining the notion of a subspace. These special sets are those that look like smaller versions of R^n that pass through the origin. These subsets have invariants called a dimension which captures a notion of size. The linear algebra definition of dimension, which uses the notion of linearly independent vectors, matches our intuition in low dimensions where lines have dimension one and planes have dimension two. These sets, and their sizes, turn out to be another tool to student matrices as functions as both the zeros and image of a matrix are subspaces of R^n.

Determinants

The determinant is a real number calculated from a square matrix that determines the invertibility of a square matrix. Its value characterizes the invertibility of the matrix. The determinant also has a geometric meaning: the absolute value of the determinant scales the volumes of sets under the function. In this module, we will show how to calculate the determinant of nxn matrices and study its properties.

Eigenvectors and Eigenvalues

In this module we study special vectors, called eigenvectors, of a linear transformation defined by a square matrix A. These are vectors whose image is easily visualized as they are scaled by a real number called the eigenvalue. While eigenvalues can be complex numbers, we do not consider that case in this course. Eigenvalues and eigenvectors are central to the theory of discrete dynamical systems, differential equations, and Markov chains and the eigentheory presented here also appear in settings in more advanced pure math courses.

Diagonalization and Linear Transformations

In this module we continue our study of eigenvalues and eigenvectors, in particular how they relate to diagonalizable matrices. Eigenvectors are so important: they make understanding linear transformations easy. They are the "axes" (directions) along which a linear transformation acts simply by "stretching/compressing" and/or "flipping"; eigenvalues give you the factors by which this compression occurs. The more directions you have along which you understand the behavior of a linear transformation, the easier it is to understand the linear transformation; so you want to have as many linearly independent eigenvectors as possible associated to a single linear transformation.

Final Assessment

Congratulations on reaching the final assessment! Review all vocabulary and theorems before attempting the final quiz below. Think about what each theorem is saying both algebraically as well as geometrically. Provide examples (with pictures in R^2 and R^3) along with counterexamples of each theorem and vocabulary term. Lastly, be sure to work through some examples for computation, looking for any of the shortcuts in the calculations when possible. In addition, there is an optional project that applies the theory of this course. You will see how eigenvalues and eigenvectors are applied to Markov Chains and the Google Page Rank algorithm. I strongly recommend you attempt this project. Good luck!

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Develops advanced techniques and theory to study matrices as linear transformations

Applies eigenvalues and eigenvectors to Markov Chains and the Google PageRank algorithm

Offers an interactive learning experience with a mix of media

Teaches linear algebra concepts that are relevant in data science, machine learning, and artificial intelligence

Provides a deeper understanding of matrix transformations through the study of eigenvalues and eigenvectors

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: Matrix Algebra, Determinants, & Eigenvectors with these activities:

Practice Matrix Algebra Operations

Show steps

Review the basic operations of matrix algebra to strengthen your understanding of matrices as linear transformations.

Browse courses on Matrix Multiplication

Show steps

Review the definitions of matrix addition, subtraction, and multiplication.
Practice performing these operations on small matrices (e.g., 2x2 and 3x3).
Verify the properties of matrix operations, such as associativity and distributivity.

Solve Systems of Linear Equations using Matrices

Show steps

Apply matrix operations to solve systems of linear equations efficiently and accurately.

Browse courses on Systems of Linear Equations

Show steps

Set up the system of equations as a matrix equation.
Perform matrix operations to transform the matrix into an upper triangular form.
Use back substitution to solve for the variables.

Attend a Linear Algebra Workshop

Show steps

Enhance your understanding of linear algebra concepts through interactive workshops led by experts in the field.

Browse courses on Linear Algebra

Show steps

Find a workshop on linear algebra that aligns with your interests.
Register for the workshop and attend all sessions.
Participate actively in discussions and ask questions to clarify concepts.

Five other activities

Expand to see all activities and additional details

Show all eight activities

Explore Geometric Properties of Matrix Transformations

Show steps

Understand the geometric significance of eigenvalues and eigenvectors in describing the behavior of matrix transformations.

Browse courses on Eigenvalues and Eigenvectors

Show steps

Follow a guided tutorial on the concepts of eigenvalues and eigenvectors.
Visualize the geometric transformations associated with different types of eigenvalues and eigenvectors.
Apply these concepts to analyze the behavior of real-world systems.

Compile Resources on Matrix Theory

Show steps

Expand your knowledge and understanding of matrix theory by gathering and organizing relevant resources.

Browse courses on Linear Algebra

Show steps

Search for books, articles, and online resources on matrix theory.
Organize the resources by topic or difficulty level.
Share the compilation with other students or interested individuals.

Identify Subspaces and Dimensionality in Matrices

Show steps

Develop a deeper understanding of subspaces and dimensionality by creating a visual representation of these concepts in matrices.

Browse courses on Subspaces

Show steps

Identify subspaces within a given matrix.
Calculate the dimensionality of these subspaces.
Create a visual representation, such as a graph or diagram, to illustrate the subspaces and their dimensionality.

Tutor Students in Linear Algebra

Show steps

Reinforce your understanding of linear algebra by helping others learn the subject and develop their problem-solving skills.

Browse courses on Linear Algebra

Show steps

Offer your services as a tutor to students in need of assistance with linear algebra.
Prepare lesson plans and materials to effectively convey concepts.
Provide individualized support and guidance to students.

Participate in a Linear Algebra Challenge

Show steps

Challenge yourself and test your understanding of linear algebra concepts by participating in a competition.

Browse courses on Linear Algebra

Show steps

Find an online or local linear algebra challenge.
Solve the problems independently or as part of a team.
Analyze your results and identify areas for improvement.

Career center

Learners who complete Linear Algebra: Matrix Algebra, Determinants, & Eigenvectors will develop knowledge and skills that may be useful to these careers:

Machine Learning Engineer

Machine Learning Engineers are responsible for designing, developing, and deploying intelligent systems. They use a variety of techniques, including machine learning and deep learning, to create models that can perform tasks such as object recognition, natural language processing, and speech recognition. Linear Algebra is important because it is used to represent data and perform operations on it. It is also used in the development of machine learning algorithms and the analysis of their performance.

See salaries and explore the career path for Machine Learning Engineer

Data Scientist

Data Scientists use their technical knowledge to solve complex problems by collecting and analyzing large amounts of data. They use their skills to build predictive models, develop algorithms, and create data visualizations. Linear Algebra is important because it is the foundation of many data science algorithms. It can be used to analyze data, build models, and predict outcomes.

See salaries and explore the career path for Data Scientist

Quantitative Analyst

Quantitative Analysts use mathematical and statistical techniques to analyze financial data and make investment decisions. They use linear algebra to solve complex financial problems, such as pricing options and forecasting stock prices. Linear algebra is also used to develop trading strategies and risk management systems.

See salaries and explore the career path for Quantitative Analyst

Operations Research Analyst

Operations Research Analysts use mathematical and statistical techniques to solve operational problems. They use linear algebra to model systems, optimize processes, and make decisions. Linear algebra is also used to develop simulation models and analyze their results.

See salaries and explore the career path for Operations Research Analyst

Actuary

Actuaries use mathematical and statistical techniques to assess risk and uncertainty. They use linear algebra to model insurance risks, calculate premiums, and develop financial plans. Linear algebra is also used to develop actuarial models and analyze their performance.

See salaries and explore the career path for Actuary

Statistician

Statisticians use mathematical and statistical techniques to collect, analyze, and interpret data. They use linear algebra to analyze data, build statistical models, and make inferences about populations. Linear algebra is also used to develop statistical methods and analyze their results.

See salaries and explore the career path for Statistician

Financial Analyst

Financial Analysts use mathematical and statistical techniques to analyze financial data and make investment recommendations. They use linear algebra to analyze financial statements, build financial models, and forecast financial performance. Linear algebra is also used to develop financial analysis tools and analyze their results.

See salaries and explore the career path for Financial Analyst

Market Researcher

Market Researchers use mathematical and statistical techniques to collect and analyze data about consumers and markets. They use linear algebra to analyze consumer behavior, design surveys, and develop marketing strategies. Linear algebra is also used to develop market research models and analyze their results.

See salaries and explore the career path for Market Researcher

Software Engineer

Software Engineers design, develop, and maintain software systems. They use linear algebra to develop algorithms, design data structures, and solve computational problems. Linear algebra is also used to develop software engineering theory and analyze its results.

See salaries and explore the career path for Software Engineer

Computer Scientist

Computer Scientists design and develop software and hardware systems. They use linear algebra to develop algorithms, design graphics, and solve computational problems. Linear algebra is also used to develop computer science theory and analyze its results.

See salaries and explore the career path for Computer Scientist

Economist

Economists use mathematical and statistical techniques to analyze economic data and make economic forecasts. They use linear algebra to model economic systems, analyze economic policies, and develop economic theories. Linear algebra is also used to develop economic models and analyze their results.

See salaries and explore the career path for Economist

Systems Engineer

Systems Engineers design, develop, and maintain complex systems. They use linear algebra to model systems, analyze system performance, and make system design decisions. Linear algebra is also used to develop systems engineering theory and analyze its results.

See salaries and explore the career path for Systems Engineer

Research Scientist

Research Scientists use mathematical and statistical techniques to conduct research and develop new technologies. They use linear algebra to develop models, analyze data, and make predictions. Linear algebra is also used to develop research methods and analyze their results.

See salaries and explore the career path for Research Scientist

Data Analyst

Data Analysts use mathematical and statistical techniques to analyze data and make data-driven decisions. They use linear algebra to analyze data, build models, and make predictions. Linear algebra is also used to develop data analytics tools and analyze their results.

See salaries and explore the career path for Data Analyst