Dave4Math / Courses / Linear Spaces

Linear Spaces

Learn about abstract linear spaces. Master the concepts of spanning sets, linear independence, bases, and finite-dimensional vector spaces. With this course, you'll see tons of examples of linear spaces and be adept at understanding the concepts and how they are related.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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Have you ever had a teacher who seemed to really enjoy some of the students in the class? What about everyone else? Introducing Dave’s online math course: Linear Spaces. Dave will help you understand every aspect of the course while being invested in every student. Your success is essential to everyone!

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in linear 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 linear spaces, linear transformations, and the matrix of a linear transformation.


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 Linear Spaces

  • Linear spaces
  • Subspaces
  • Span, linear independence, basis, coordinates
  • Finite dimensional linear spaces
  • Linear Transformations, Image, Kernel, Rank, Nullity
  • Definitions and Examples of Span and Linear Independence
  • Span Is the Smallest Containing Subspace
  • More on Linear Independence
  • Linear Dependence Lemma
  • Finite-Dimensional Subspaces
  • Introduction to Bases
  • Criterion for Basis
  • Spanning List Contains a Basis
  • Basis of Finite-Dimensional Vector Space
  • Linearly Independent List Extends to a Basis
  • Introduction to Dimension
  • Basis Length
  • Dimension of a Subspace
  • Linearly Independent List of the Right Length Is a Basis
  • Spanning List of the Right Length Is a Basis
  • Dimension of a Sum
  • Isomorphisms and Isomorphic Spaces
  • Properties of Isomorphisms
  • The B-matrix of a Linear Transformation
  • Change of Basis

Course Description

In the next lesson, we study linear transformations with a special emphasis on isomorphisms. In particular, we detail the image, kernel, rank, and nullity of a linear transformation. Many properties of isomorphism are covered, and strategies for determining isomorphism are detailed.

At the end of the course, we illustrate through several examples the matrix of a linear transformation. The change of basis matrix and similar matrices are also featured.

Recommended Prerequisites for Linear Spaces

I recommend the prerequisite course Subspaces and Their Dimension. 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.

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.