Dave4Math / Courses / Orthogonality


Master orthogonality and inner product spaces. Learn about orthogonal projections, the Gram–Schmidt process, and orthogonal transformations. With this course, you'll understand the importance of orthogonality and become experienced with inner product spaces.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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Does your teacher only look at you whenever they want something from you, like the validation they teach? So they want an answer to their question on the spot in the middle of class. Introducing Dave’s online math course: Orthogonality. Dave is well-known for being an engaging teacher and an effective communicator. He cares about your achievements!

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in orthogonality.
  • Students who are enrolled in linear algebra and want to improve their grade.
  • Anyone interested in majoring in mathematics, physics, or engineering.
  • Anyone wanting to learn about orthogonal projections, orthonormal bases, Gram-Schmidt process, QR- factorization, orthogonal transformations, and orthogonal matrices.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is recommended.

What You’ll Learn in Orthogonality

  • Orthogonality, Length, Unit Vectors
  • Orthonormal Vectors
  • Orthogonal Projections
  • Orthogonal Complement
  • From Pythagoras to Cauchy
  • The Gram–Schmidt Process
  • The QR Factorization
  • Orthogonal Transformations
  • Orthogonal Matrices
  • Orthonormal Bases
  • The Transpose of a Matrix
  • The Matrix of an Orthogonal Projection
  • Characterization of Orthogonal Complements
  • Characterization of Orthogonal Projections
  • Least-Squares Approximations
  • The Matrix of an Orthogonal Projection
  • Data Fitting
  • Inner Products and Inner Product Spaces
  • Norm, Orthogonality
  • Orthogonal Projections
  • Fourier Analysis

Course Description

We begin this course by studying orthogonal vectors, the length of a vector, and unit vectors. This introduction leads us to an understanding of orthonormal vectors, including some of their properties. Next, we dive into orthogonal projections, including what they are and how to find them. We spend a great deal of attention on the formula for finding the orthogonal projection and also on the orthogonal complement. After that, we generalize several well-known theorems of real vector spaces, including the Pythagorean Theorem, the Cauchy-Schwarz inequality, and the Law of Cosines.

Next, we study one of the main topics of Linear Algebra, the Gram-Schmidt Process. We demonstrate this process with several examples, and we provide proof of this fundamental result. Now because the Gram-Schmidt process represents a change of basis (from a basis to an orthonormal basis), it is most easily described in terms of a change of basis matrix. Hence, an effective way to organize and record the work performed in the Gram-Schmidt process is via QR-factorization.

Next, we learn that a linear transformation that preserves the length of vectors is called an orthogonal transformation. Interestingly, orthogonal transformations preserve orthogonality. We give a detailed study of orthogonal transformations and their products and inverses. Towards the end of this lesson, we summarize orthogonal matrices and properties of the transpose.

The final lesson of this course is for studying inner product spaces. We define what an inner product is, what an inner product space is, and provide several examples. Then we cover the norm and orthogonal projections before giving a quick view of Fourier analysis, which includes Fourier coefficients and the Fourier approximation.

Recommended Prerequisites for Orthogonality

I recommend the prerequisite course Linear Spaces. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free articles on linear algebra.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.