Johns Hopkins University
Linear Algebra: Orthogonality and Diagonalization
Johns Hopkins University

Linear Algebra: Orthogonality and Diagonalization

This course is part of Linear Algebra from Elementary to Advanced Specialization

Taught in English

Joseph W. Cutrone, PhD

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Course

Gain insight into a topic and learn the fundamentals

4.9

(12 reviews)

Intermediate level
Some related experience required
9 hours (approximately)
Flexible schedule
Learn at your own pace

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Assessments

11 quizzes

Course

Gain insight into a topic and learn the fundamentals

4.9

(12 reviews)

Intermediate level
Some related experience required
9 hours (approximately)
Flexible schedule
Learn at your own pace

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This course is part of the Linear Algebra from Elementary to Advanced Specialization
When you enroll in this course, you'll also be enrolled in this Specialization.
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There are 4 modules in this course

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.

What's included

2 videos2 readings3 quizzes

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.

What's included

3 videos3 readings4 quizzes

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.

What's included

2 videos2 readings3 quizzes

What's included

1 quiz

Instructor

Joseph W. Cutrone, PhD

Top Instructor

Johns Hopkins University
19 Courses395,213 learners

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