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Joseph W. Cutrone, PhD
This course is the second course in the Linear Algebra Specialization. In this course, we continue to develop the techniques and theory to study matrices as special linear transformations (functions) on vectors. In particular, we develop techniques to manipulate matrices algebraically. This will allow us to better analyze and solve systems of linear equations. Furthermore, the definitions and theorems presented in the course allow use to identify the properties of an invertible matrix, identify relevant subspaces in R^n,
Read more
This course is the second course in the Linear Algebra Specialization. In this course, we continue to develop the techniques and theory to study matrices as special linear transformations (functions) on vectors. In particular, we develop techniques to manipulate matrices algebraically. This will allow us to better analyze and solve systems of linear equations. Furthermore, the definitions and theorems presented in the course allow use to identify the properties of an invertible matrix, identify relevant subspaces in R^n,
Read more
This course is the second course in the Linear Algebra Specialization. In this course, we continue to develop the techniques and theory to study matrices as special linear transformations (functions) on vectors. In particular, we develop techniques to manipulate matrices algebraically. This will allow us to better analyze and solve systems of linear equations. Furthermore, the definitions and theorems presented in the course allow use to identify the properties of an invertible matrix, identify relevant subspaces in R^n,
We then focus on the geometry of the matrix transformation by studying the eigenvalues and eigenvectors of matrices. These numbers are useful for both pure and applied concepts in mathematics, data science, machine learning, artificial intelligence, and dynamical systems. We will see an application of Markov Chains and the Google PageRank Algorithm at the end of the course.
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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.
Read more
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.
Read more
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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
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Reviews summary
Comprehensive matrix algebra and eigenvector fundamentals
According to learners, this course offers a strong and comprehensive understanding of linear algebra, particularly in matrix algebra, determinants, and eigenvectors. Many praise the instructor's exceptional clarity in explaining complex topics, making them approachable and intuitive. Students frequently highlight the well-structured modules, which break down concepts effectively. While generally lauded for its depth, some older reviews indicate that the pace might be challenging for absolute beginners or those without prior mathematical exposure, suggesting a need for supplementary practice. The optional project on Markov Chains and Google PageRank is consistently cited as a valuable application of theory, connecting abstract concepts to real-world scenarios in data science, machine learning, and AI. Overall, it's considered a highly recommended course for its practical relevance and detailed instruction.
Beneficial for those with some math background; can be fast-paced for true beginners.
"I found the pace quite fast at times, especially in the Determinants section; prior exposure to calculus or discrete math helps."
"This course assumes a certain level of mathematical maturity; some sections felt quite intense without a strong pre-existing foundation."
"While excellent, I think this course is best suited for learners who are comfortable with mathematical notation and abstract thinking."
The course offers an engaging and often intuitive learning experience.
"I genuinely enjoyed the lectures; they kept me engaged and curious about the next concept."
"The examples provided in the lectures were spot on, helping me visualize complex ideas effectively."
"I appreciated the consistent encouragement and clear passion for the subject from the instructor throughout the course."
The course is well-organized, covering core topics with appropriate depth.
"The content is very well-structured, moving logically from basic matrix operations to more advanced eigenvector concepts."
"I found the modules to be perfectly paced, building on previous knowledge without overwhelming me."
"This course provides a solid foundation, especially on subspaces and determinants, which I previously struggled with."
Strong emphasis on real-world applications, particularly for ML/AI.
"The Markov Chains and Google PageRank project was incredibly valuable for understanding the practical side of linear algebra."
"I enjoyed seeing how the theoretical concepts applied directly to machine learning algorithms, which was a big draw for me."
"The discussions on the use of eigenvalues in data science made the abstract concepts feel much more relevant and engaging."
The instructor's explanations are consistently clear and highly effective.
"The instructor is exceptional; I've taken many math courses, and this is by far the best in terms of explanations."
"Dr. X really knows how to break down complex topics into easily digestible parts, making linear algebra much less intimidating."
"I appreciated how the instructor clarified common pitfalls and provided intuition behind abstract concepts."
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.
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
Show steps
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
Show steps
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
Show steps
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
Show steps
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
Show steps
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
Show steps
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
Show steps
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
Show steps
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 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.
Data Scientist
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.
Quantitative Analyst
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.
Actuary
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.
Operations Research 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.
Statistician
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.
Market Researcher
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.
Financial Analyst
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.
Computer Scientist
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.
Economist
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.
Software Engineer
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.
Data Analyst
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.
Research Scientist
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.
Systems Engineer
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.
For more career information including salaries, visit:
OpenCourser.com/course/3vpydz/linear
Reading list
We've selected ten 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: Matrix Algebra, Determinants, & Eigenvectors.
Provides a comprehensive introduction to linear algebra, covering the same topics as the course. It classic textbook that is widely used in undergraduate courses and is known for its clear explanations and examples.
Popular textbook that provides a clear and concise introduction to linear algebra. It is known for its friendly writing style and its use of real-world examples to illustrate concepts.
Save
Comprehensive reference on matrix analysis. It covers a wide range of topics, including matrix theory, matrix computations, and matrix applications. It valuable resource for anyone who wants to learn more about matrices.
Save
Provides a comprehensive introduction to matrix computations. It covers a wide range of topics, including matrix theory, matrix computations, and matrix applications. It valuable resource for anyone who wants to learn more about matrices.
Provides a comprehensive introduction to linear algebra and optimization with a focus on applications in machine learning. It is known for its clear and concise writing style and its use of examples to illustrate concepts.
Provides a comprehensive introduction to numerical linear algebra. It covers a wide range of topics, including matrix theory, matrix computations, and matrix applications. It valuable resource for anyone who wants to learn more about numerical linear algebra.
Provides a modern and accessible introduction to linear algebra. It is known for its focus on applications and its use of technology to enhance the learning experience.
More advanced textbook that provides a rigorous treatment of linear algebra. It is known for its clear and concise writing style and its focus on proofs.
Provides a comprehensive introduction to linear algebra with a focus on applications to differential equations. It is known for its clear and concise writing style and its use of examples to illustrate concepts.
Provides a concise and accessible introduction to linear algebra. It is known for its focus on the essential concepts of linear algebra and its use of examples to illustrate those concepts.
For more information about how these books relate to this course, visit:
OpenCourser.com/course/3vpydz/linear
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Via
Coursera
Institution
Johns Hopkins University
Instructor
Joseph W. Cutrone, PhD
Language
English
This course belongs to a
collection
called
Linear Algebra from Elementary to Advanced. It's comprised of three courses:
- Linear Algebra: Linear Systems and Matrix Equations
- Linear Algebra: Orthogonality and Diagonalization
- Linear Algebra: Matrix Algebra, Determinants, & Eigenvectors (viewing)
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