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Linear Algebra

Orthogonality and Diagonalization

Joseph W. Cutrone, PhD

This is the third and final course in the Linear Algebra Specialization that focuses on the theory and computations that arise from working with orthogonal vectors. This includes the study of orthogonal transformation, orthogonal bases, and orthogonal transformations. The course culminates in the theory of symmetric matrices, linking the algebraic properties with their corresponding geometric equivalences. These matrices arise more often in applications than any other class of matrices.

The theory, skills and techniques learned in this course have applications to AI and machine learning. In these popular fields, often the driving engine behind the systems that are interpreting, training, and using external data is exactly the matrix analysis arising from the content in this course.

Successful completion of this specialization will prepare students to take advanced courses in data science, AI, and mathematics.

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Syllabus

Orthogonality

In this module, we define a new operation on vectors called the dot product. This operation is a function that returns a scalar related to the angle between the vectors, distance between vectors, and length of vectors. After working through the theory and examples, we hone in on both unit (length one) and orthogonal (perpendicular) vectors. These special vectors will be pivotal in our course as we start to define linear transformations and special matrices that use only these vectors.

Orthogonal Projections and Least Squares Problems

In this module we will study the special type of transformation called the orthogonal projection. We have already seen the formula for the orthogonal projection onto a line so now we generalize the formula to the case of projection onto any subspace W. The formula will require basis vectors that are both orthogonal and normalize and we show, using the Gram-Schmidt Process, how to meet these requirements given any non-empty basis.

Symmetric Matrices and Quadratic Forms

In this module we look to diagonalize symmetric matrices. The symmetry displayed in the matrix A turns out to force a beautiful relationship between the eigenspaces. The corresponding eigenspaces turn out to be mutually orthogonal. After normalizing, these orthogonal eigenvectors give a very special basis of R^n with extremely useful applications to data science, machine learning, and image processing. We introduce the notion of quadratic forms, special functions of degree two on vectors , which use symmetric matrices in their definition. Quadratic forms are then completely classified based on the properties of their eigenvalues.

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Examines linear algebra concepts such as orthogonality and symmetric matrices, which have practical applications in AI and machine learning

Taught by Joseph W. Cutrone, PhD, a seasoned professional in the field

Suitable for advanced learners in data science, AI, and mathematics

Builds a solid foundation in linear algebra, preparing learners for further study and practical applications

Involves hands-on exercises and real-world examples to enhance understanding

Might require familiarity with basic linear algebra concepts for optimal comprehension

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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: Orthogonality and Diagonalization with these activities:

Review Linear Algebra Basics

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Builds a strong foundation for understanding the more advanced concepts covered in this course.

Browse courses on Vectors

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Review notes from previous linear algebra courses.
Go over online resources or textbooks to refresh your memory.
Complete practice problems to test your understanding.

Form a Study Group

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Facilitates collaboration, encourages diverse perspectives, and provides a supportive learning environment.

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Find classmates who are interested in forming a study group.
Set regular meeting times and locations.
Establish clear goals and expectations.
Take turns leading discussions and presenting material.

Read Linear Algebra: A Modern Introduction

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Provides a solid theoretical foundation and practical applications for understanding the concepts covered in this course.

View Student Solutions Manual for Poole's Linear... on Amazon

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Read the first chapter to gain an overview of the subject.
Complete the practice problems at the end of each chapter to test your understanding.
Meet with a study group to discuss the concepts.

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Watch Video Lectures on Orthogonal Matrices

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Provides visual explanations and demonstrations to enhance understanding.

Browse courses on Orthogonal Matrices

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Search for video lectures on orthogonal matrices.
Take notes while watching the lectures.
Pause the videos and rewind as needed to fully grasp the concepts.

Solve Linear Algebra Problems

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Reinforces the concepts and techniques learned in the course through hands-on problem-solving.

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Find practice problems online or in textbooks.
Set aside dedicated time to practice solving problems.
Check your solutions and identify areas for improvement.

Develop a Presentation on Symmetric Matrices

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Encourages deep understanding and the ability to communicate complex concepts effectively.

Browse courses on Symmetric Matrices

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Research the topic thoroughly.
Create a clear and engaging presentation.
Practice delivering the presentation.
Present the project to your classmates or a wider audience.

Contribute to an Open-Source Linear Algebra Library

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Provides practical experience with linear algebra concepts while contributing to the community.

Browse courses on Open Source

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Find an open-source linear algebra library that you're interested in.
Review the codebase and identify an area where you can contribute.
Implement your changes and submit a pull request.
Collaborate with other contributors to refine your code and get it merged.

Participate in a Math Contest

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Provides a challenging environment to test skills and push the limits of understanding.

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Research math contests and identify one that aligns with your interests.
Register for the contest and prepare thoroughly.
Compete in the contest and give it your best effort.
Reflect on your performance and identify areas for growth.

Career center

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

Data Scientist

Data Scientists use advanced mathematical techniques and programming skills to extract valuable knowledge from data. This course provides a strong foundation in the linear algebra concepts that are essential for data scientists, such as orthogonality, projection, and diagonalization. These concepts are used in various data science applications, such as dimensionality reduction, feature selection, and machine learning algorithms. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind data science and improve their ability to apply these principles to real-world data analysis problems.

See salaries and explore the career path for Data Scientist

Machine Learning Engineer

Machine Learning Engineers design, develop, and maintain machine learning models that can learn from data and make predictions. This course provides a solid understanding of the linear algebra concepts that underpin machine learning algorithms, such as orthogonality, projection, and diagonalization. These concepts are essential for understanding the behavior of machine learning models and for developing new and improved algorithms. By taking this course, learners can gain a deeper understanding of the mathematical foundations of machine learning and enhance their ability to develop and deploy effective machine learning solutions.

See salaries and explore the career path for Machine Learning Engineer

Quantitative Analyst

Quantitative Analysts use mathematical and statistical models to analyze financial data and make investment decisions. This course provides a strong foundation in the linear algebra concepts that are essential for quantitative analysts, such as orthogonality, projection, and diagonalization. These concepts are used in various financial applications, such as portfolio optimization, risk management, and trading strategies. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind quantitative finance and improve their ability to apply these principles to real-world investment decisions.

See salaries and explore the career path for Quantitative Analyst

Actuary

Actuaries use mathematical and statistical techniques to assess and manage risk. This course provides a strong foundation in the linear algebra concepts that are essential for actuaries, such as orthogonality, projection, and diagonalization. These concepts are used in various actuarial applications, such as pricing insurance policies, calculating reserves, and managing risk portfolios. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind actuarial science and improve their ability to apply these principles to real-world risk management problems.

See salaries and explore the career path for Actuary

Operations Research Analyst

Operations Research Analysts use mathematical and analytical techniques to solve complex business problems. This course provides a strong foundation in the linear algebra concepts that are essential for operations research analysts, such as orthogonality, projection, and diagonalization. These concepts are used in various operations research applications, such as supply chain management, logistics, and scheduling. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind operations research and improve their ability to apply these principles to real-world business problems.

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 the linear algebra concepts that are essential for statisticians, such as orthogonality, projection, and diagonalization. These concepts are used in various statistical applications, such as regression analysis, hypothesis testing, and data visualization. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind statistics and improve their ability to apply these principles to real-world data analysis problems.

See salaries and explore the career path for Statistician

Financial Analyst

Financial Analysts use mathematical and financial techniques to analyze and make recommendations on investments. This course provides a strong foundation in the linear algebra concepts that are essential for financial analysts, such as orthogonality, projection, and diagonalization. These concepts are used in various financial applications, such as portfolio management, security analysis, and risk assessment. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind financial analysis and improve their ability to apply these principles to real-world investment decisions.

See salaries and explore the career path for Financial Analyst

Risk Manager

Risk Managers identify, assess, and manage risks to an organization. This course provides a strong foundation in the linear algebra concepts that are essential for risk managers, such as orthogonality, projection, and diagonalization. These concepts are used in various risk management applications, such as risk assessment, risk modeling, and risk mitigation. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind risk management and improve their ability to apply these principles to real-world risk management problems.

See salaries and explore the career path for Risk Manager

Investment Analyst

Investment Analysts research and analyze potential investments to make recommendations to clients. This course provides a strong foundation in the linear algebra concepts that are essential for investment analysts, such as orthogonality, projection, and diagonalization. These concepts are used in various investment applications, such as portfolio management, security analysis, and risk assessment. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind investment analysis and improve their ability to apply these principles to real-world investment decisions.

See salaries and explore the career path for Investment Analyst

Business Analyst

Business Analysts use analytical techniques to solve business problems and improve organizational performance. This course provides a strong foundation in the linear algebra concepts that are essential for business analysts, such as orthogonality, projection, and diagonalization. These concepts are used in various business analysis applications, such as data analysis, process improvement, and risk management. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind business analysis and improve their ability to apply these principles to real-world business problems.

See salaries and explore the career path for Business Analyst

Data Analyst

Data Analysts collect, clean, and analyze data to identify trends and patterns. This course provides a strong foundation in the linear algebra concepts that are essential for data analysts, such as orthogonality, projection, and diagonalization. These concepts are used in various data analysis applications, such as data mining, machine learning, and statistical modeling. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind data analysis and improve their ability to apply these principles to real-world data analysis problems.

See salaries and explore the career path for Data Analyst

Software Engineer

Software Engineers design, develop, and maintain software systems. This course provides a strong foundation in the linear algebra concepts that are essential for software engineers, such as orthogonality, projection, and diagonalization. These concepts are used in various software engineering applications, such as computer graphics, image processing, and scientific computing. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind software engineering and improve their ability to design and develop effective software systems.

See salaries and explore the career path for Software Engineer

Computer Scientist

Computer Scientists research and develop new computing technologies. This course provides a strong foundation in the linear algebra concepts that are essential for computer scientists, such as orthogonality, projection, and diagonalization. These concepts are used in various computer science applications, such as artificial intelligence, computer graphics, and scientific computing. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind computer science and improve their ability to research and develop new computing technologies.

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Mathematician

Mathematicians research and develop new mathematical theories and techniques. This course provides a strong foundation in the linear algebra concepts that are essential for mathematicians, such as orthogonality, projection, and diagonalization. These concepts are used in various mathematical applications, such as algebra, geometry, and analysis. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind mathematics and improve their ability to research and develop new mathematical theories and techniques.

See salaries and explore the career path for Mathematician

Physicist

Physicists research and develop new theories and models to explain the physical world. This course provides a strong foundation in the linear algebra concepts that are essential for physicists, such as orthogonality, projection, and diagonalization. These concepts are used in various physics applications, such as quantum mechanics, electromagnetism, and thermodynamics. By taking this course, learners can enhance their understanding of the underlying mathematical principles behind physics and improve their ability to research and develop new physical theories and models.

See salaries and explore the career path for Physicist

Reading list

We've selected 11 books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Linear Algebra: Orthogonality and Diagonalization.

Linear Algebra Done Right

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Valuable resource for students and professionals who want to learn more about linear algebra. The author provides a comprehensive and theoretical coverage of the subject matter.

Linear Algebra Done Right (Undergraduate Texts in...

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Linear Algebra Done Right (Undergraduate Texts in...

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Linear Algebra and Its Applications

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Is an excellent resource for students and professionals who want to learn more about linear algebra. The author makes the subject accessible to readers with a variety of backgrounds.

Linear Algebra

Orthogonality and Diagonalization

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