Linear Algebra IV: Orthogonality & Symmetric Matrices and the SVD from edX

In the first part of this course you will explore methods to compute an approximate solution to an inconsistent system of equations that have no solutions. Our overall approach is to center our algorithms on the concept of distance. To this end, you will first tackle the ideas of distance and orthogonality in a vector space. You will then apply orthogonality to identify the point within a subspace that is nearest to a point outside of it. This has a central role in the understanding of solutions to inconsistent systems. By taking the subspace to be the column space of a matrix, you will develop a method for producing approximate (“least-squares”) solutions for inconsistent systems.

You will then explore another application of orthogonal projections: creating a matrix factorization widely used in practical applications of linear algebra. The remaining sections examine some of the many least-squares problems that arise in applications, including the least squares procedure with more general polynomials and functions.

This course then turns to symmetric matrices. arise more often in applications, in one way or another, than any other major class of matrices. You will construct the diagonalization of a symmetric matrix, which gives a basis for the remainder of the course.

What you'll learn

Upon completion of this course, learners will be able to:

Compute dot product of two vectors, length of a vector, distance between points, and angles between vectors
Apply theorems related to orthogonal complements, and their relationships to Row and Null
space, to characterize vectors and linear systems
Compute orthogonal projections and distances to express a vector as a linear combination of orthogonal vectors, construct vector approximations using projections, and characterize bases for subspaces, and construct orthonormal bases
Apply the iterative Gram Schmidt Process, and the QR decomposition, to construct an orthogonal basis
Construct the QR factorization of a matrix
Characterize properties of a matrix using its QR decomposition
Compute general solutions and least squares errors to least squares problems using the normal
equations and the QR decomposition
Apply least-squares and multiple regression to construct a linear model from a set of data
points
Apply least-squares to fit polynomials and other curves to data
Construct an orthogonal diagonalization of a symmetric matrix
Construct a spectral decomposition of a matrix

What's inside

Learning objectives

Compute dot product of two vectors, length of a vector, distance between points, and angles between vectors
Apply theorems related to orthogonal complements, and their relationships to row and null
space, to characterize vectors and linear systems
Compute orthogonal projections and distances to express a vector as a linear combination of orthogonal vectors, construct vector approximations using projections, and characterize bases for subspaces, and construct orthonormal bases
Apply the iterative gram schmidt process, and the qr decomposition, to construct an orthogonal basis
Construct the qr factorization of a matrix
Characterize properties of a matrix using its qr decomposition

Compute general solutions and least squares errors to least squares problems using the normal
equations and the qr decomposition
Apply least-squares and multiple regression to construct a linear model from a set of data
points
Apply least-squares to fit polynomials and other curves to data
Construct an orthogonal diagonalization of a symmetric matrix
Construct a spectral decomposition of a matrix

Compute dot product of two vectors, length of a vector, distance between points, and angles between vectors
Apply theorems related to orthogonal complements, and their relationships to row and null
space, to characterize vectors and linear systems
Compute orthogonal projections and distances to express a vector as a linear combination of orthogonal vectors, construct vector approximations using projections, and characterize bases for subspaces, and construct orthonormal bases
Apply the iterative gram schmidt process, and the qr decomposition, to construct an orthogonal basis
Construct the qr factorization of a matrix
Characterize properties of a matrix using its qr decomposition
Compute general solutions and least squares errors to least squares problems using the normal
equations and the qr decomposition
Apply least-squares and multiple regression to construct a linear model from a set of data
points
Apply least-squares to fit polynomials and other curves to data
Construct an orthogonal diagonalization of a symmetric matrix
Construct a spectral decomposition of a matrix

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Emphasizes distance as a cornerstone for identifying approximate solutions to inconsistent linear systems

Introduces the concept of orthogonal projections to tackle inconsistent systems and construct the QR factorization

Provides a comprehensive understanding of orthogonal complements and their significance in linear systems

Builds on the concept of least-squares problems and their applications in various domains

Delves into symmetric matrices, a type commonly encountered in practical applications of linear algebra

Reviews summary

In-depth linear algebra

According to students, this course presents basic but innovative mathematical analysis in linear algebra. Students appreciated the dedication of the professors who helped to improve students' talents, activities, personal attitudes, and mental fitness in addition to their understanding of the subject.

Instructors are dedicated.

"When the course period the professors gave a perfect and dedication of their knowledge to improve improve the students talent ,activities,personal attitude,mentally fit ,bold ,talkative easily with others for subject basics to current affairs."

This course is in-depth and engaging.

"The course contains basic ,innovative applied mathematical analysis."

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 IV: Orthogonality & Symmetric Matrices and the SVD with these activities:

Read Gilbert Strang's 'Introduction to Linear Algebra'

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Reading Strang's textbook will provide you with a comprehensive understanding of linear algebra, including the concepts covered in this course.

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Read the assigned chapters
Work through the practice problems

Organize Course Materials

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By organizing your notes, assignments, and quizzes, you can improve your understanding of the material and prepare for assessments more effectively.

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Review and summarize lecture notes
Categorize and file assignments
Organize quizzes and practice problems

Practice Matrix Calculations

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Practicing matrix calculations will improve your understanding of the concepts covered in this course and enhance your problem-solving skills.

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Find the determinant of a matrix
Perform matrix multiplication
Compute the inverse of a matrix

Three other activities

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Solve Least Squares Problems

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Practicing solving least squares problems will enhance your ability to use this technique in real-world applications, such as data fitting and parameter estimation.

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Formulate the least squares problem
Use the normal equations or QR decomposition to solve the problem
Interpret the results and assess the goodness of fit

Develop a QR Factorization Implementation

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Implementing the QR factorization algorithm will solidify your understanding of the technique and its applications in solving systems of equations and least squares problems.

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Write the algorithm in a programming language
Test the implementation on various matrices
Analyze the results and identify any potential improvements

Apply Linear Algebra to Real-World Scenarios

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Solving real-world problems using linear algebra will enhance your problem-solving skills and demonstrate the practical applications of the concepts covered in this course.

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Identify a real-world problem that can be modeled using linear algebra
Formulate the problem as a system of linear equations or matrices
Solve the problem using techniques learned in the course

Career center

Learners who complete Linear Algebra IV: Orthogonality & Symmetric Matrices and the SVD will develop knowledge and skills that may be useful to these careers:

Statistician

Statisticians leverage their linear algebra skills to model and analyze data. This course's focus on orthogonal projections and the Singular Value Decomposition (SVD) provides a strong foundation for understanding the underlying principles of statistical modeling. The course's exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for real-world scenarios in statistics.

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Data Scientist

Data scientists rely heavily on linear algebra for data analysis and modeling. This course's in-depth exploration of orthogonal projections, the QR decomposition, and the SVD provides a solid foundation for understanding the techniques used in data science. The course's emphasis on least-squares problems and applications in multiple regression and curve fitting directly translates to practical tasks in data science.

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Machine Learning Engineer

Machine learning engineers apply linear algebra to develop and optimize machine learning algorithms. This course's coverage of orthogonal projections and the SVD provides a strong understanding of the underlying mathematical concepts. The exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for tasks involving model fitting and optimization.

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Financial Analyst

Financial analysts utilize linear algebra for risk assessment and portfolio optimization. This course's focus on orthogonal projections and the SVD provides a solid foundation for understanding the principles of financial modeling. The course's exploration of least-squares problems and applications in multiple regression and curve fitting directly translates to practical tasks in financial analysis.

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Actuary

Actuaries apply linear algebra to assess risk and develop insurance products. This course's coverage of orthogonal projections and the SVD provides a strong foundation for understanding the mathematical principles behind actuarial models. The exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for tasks involving risk assessment and premium pricing.

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Operations Research Analyst

Operations research analysts leverage linear algebra for optimization and decision-making. This course's focus on orthogonal projections and the SVD provides a strong understanding of the underlying mathematical principles. The exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for tasks involving resource allocation and process optimization.

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Software Engineer

Software engineers utilize linear algebra for computer graphics, data compression, and optimization. This course's coverage of orthogonal projections and the SVD provides a solid foundation for understanding the principles behind these applications. The exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for tasks involving image processing and algorithm optimization.

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Quantitative Analyst

Quantitative analysts employ linear algebra for risk management and trading strategies. This course's focus on orthogonal projections and the SVD provides a strong foundation for understanding the principles behind financial modeling. The exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for tasks involving portfolio optimization and risk assessment.

See salaries and explore the career path for Quantitative Analyst

Market Researcher

Market researchers leverage linear algebra for data analysis and consumer behavior modeling. This course's focus on orthogonal projections and the SVD provides a strong foundation for understanding the underlying principles. The exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for tasks involving market segmentation and forecasting.

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Economist

Economists use linear algebra for economic modeling and forecasting. This course's coverage of orthogonal projections and the SVD provides a solid foundation for understanding the principles behind economic models. The exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for tasks involving economic analysis and forecasting.

See salaries and explore the career path for Economist

Financial Risk Manager

Financial risk managers utilize linear algebra for risk assessment and portfolio optimization. This course's focus on orthogonal projections and the SVD provides a strong foundation for understanding the principles behind risk models. The exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for tasks involving risk management and capital allocation.

See salaries and explore the career path for Financial Risk Manager

Biostatistician

Biostatisticians apply linear algebra for medical research and data analysis. This course's focus on orthogonal projections and the SVD provides a strong foundation for understanding the principles behind statistical modeling in biology and medicine. The exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for tasks involving clinical trial design and analysis.

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Geophysicist

Geophysicists utilize linear algebra for data analysis and modeling of Earth's structure and processes. This course's focus on orthogonal projections and the SVD provides a strong foundation for understanding the underlying principles. The exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for tasks involving seismic imaging and resource exploration.

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Computer Scientist

Computer scientists leverage linear algebra for computer graphics, image processing, and data mining. This course's coverage of orthogonal projections and the SVD provides a solid foundation for understanding the principles behind these applications. The exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for tasks involving machine learning and data analysis.

See salaries and explore the career path for Computer Scientist

Aerospace Engineer

Aerospace engineers employ linear algebra for aircraft design and flight mechanics. This course's focus on orthogonal projections and the SVD provides a strong foundation for understanding the principles behind aerodynamic modeling. The exploration of least-squares problems and applications in multiple regression and curve fitting further prepares learners for tasks involving flight simulation and control.

See salaries and explore the career path for Aerospace Engineer