Dave4Math / Courses / Operators on Vector Spaces

Operators on Vector Spaces

Master operators on vector spaces, including real spaces, complex spaces, and inner-product spaces. Learn the structure of a linear operator. With this course, you will be proficient at working with linear operators and finding their structure.


January 26, 2021

Created by Dave

This course begins Summer 2021.
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Do you wish your linear algebra class went further? This excellent course is an advanced course for beginners who want to understand everything in linear algebra. Introducing Dave’s online math course: Operators on Vector Spaces. Dave makes this subject easy to understand and explains it clearly. You will be challenged and yet enjoy it!

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in operators on vector spaces.
  • 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 invariant subspaces, operators on inner product spaces, operators on complex vector spaces, operators on real vector spaces, and operators on real inner product spaces.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator is not necessary.

What You’ll Learn in Operators on Vector Spaces

  • Invariant Subspaces
  • Eigenvectors
  • Upper-Triangular Matrices
  • Eigenspaces
  • Diagonal Matrices
  • Operators on Inner Product Spaces
  • Self-Adjoint Operators
  • Normal Operators
  • The Spectral Theorem
  • Positive Operators
  • Isometries
  • Polar Decomposition
  • Singular Value Decomposition
  • Operators on Complex Vector Spaces
  • Generalized Eigenvectors
  • Nilpotent Operators
  • Generalized Eigenvectors
  • Nilpotent Operators
  • Characteristic Polynomial
  • Cayley-Hamilton Theorem
  • Decomposition of an Operator
  • Minimal Polynomial
  • Jordan Form
  • Operators on Real Vector Spaces
  • Complexification of a Vector Space
  • Complexification of an Operator
  • The Minimal Polynomial of the Complexification
  • Eigenvalues of the Complexification
  • Characteristic Polynomial of the Complexification
  • Normal Operators on Real Inner Product Spaces
  • Isometries on Real Inner Product Spaces

Course Description

This advanced course is a continuation of these courses: Linear Transformations, Linear Spaces, Subspaces and Their Dimension, Orthogonality, and Eigenvalues and Eigenvectors.

In this course, we begin by studying operators on inner product spaces. First, we discuss self-adjoint and normal operators. Then we give an in-depth examination of the Spectral Theorem, both the complex and the real version. We finish this first lesson by detailing positive operators and isometries.

In the next lesson, we study the main topic of this course: operators on complex vector spaces. To do so, we start off by learning about generalized eigenvectors and nilpotent operators. We then work through many of the details and provide lots of examples. Now, we come to the heart of the matter: the decomposition of an operator. The details here are essential, and we work thoroughly on the material, including the theory of characteristic and minimal polynomials. After that, we celebrate the Cayley-Hamilton Theorem and then end with the Jordan Form of a matrix.

In the final lesson, we begin with the complexification of a real vector space. We go through this process and the resulting properties in detail before continuing on with operators of real inner product spaces. We end this course with a discussion on isometries on real inner product spaces.

Recommended Prerequisites for Operators on Vector Spaces

I recommend the prerequisite course Eigenvalues and Eigenvectors. 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.