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Subspaces and Their Dimension

Master subspaces and their dimension. In these lessons, you'll learn about subspaces, spanning sets, and the kernel and image. With this course, you'll become knowledgeable about subspaces and why they're useful and skilled at finding their dimension.


January 11, 2021

Created by Dave

This course begins Summer 2021.
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Attitude. Do you have it? That demeanor of confidence in your knowledge and skill helps you accomplish. Introducing Dave’s online math course: Subspaces and Their Dimension. Dave goes into each lesson with a fun attitude and keeps the material enjoyable and easier to learn. You can have math confidence too!

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in subspaces and their dimension.
  • 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 the image and kernel of a linear transformation, subspaces, bases, linear independence, the dimension of a subspace, and coordinates.


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 Subspaces and Their Dimension

  • The Image of a Linear Transformation
  • The Kernel of a Linear Transformation
  • Characterizations of Invertible Matrices
  • Subspaces
  • Bases and Linear Independence
  • Characterizations of Linear Independence
  • The Dimension of a Subspace
  • Number of Vectors in a Basis
  • Finding Bases of Kernel and Image
  • Rank-Nullity Theorem
  • Introduction to Coordinates
  • Linearity of Coordinates
  • The Matrix of a Linear Transformation
  • Similar Matrices

Course Description

This course is a continuation of Linear Transformations. So we begin with a thorough introduction to the image and kernel of a linear transformation. Then we discuss spanning sets and properties of the image of a linear transformation. Properties of the kernel are also detailed, and importantly, various characterization of invertible matrices are given.

By considering the properties of the image and kernel of a linear transformation, we motivate the definition of a subspace of a vector space. After that, we prove that the kernel and image are subspaces, and then we provide several examples of subspaces.

By working through examples, the idea of redundant vectors comes into focus. So we thoroughly examine the concepts of linear independence and bases. Then we cover linear relations and provide several characterizations of linear independence.

Finally, in this course, we study the dimension of a subspace. We prove that every basis for a subspace has the same number of vectors (called the dimension of the subspace). Many examples are given, and we demonstrate in great detail finding a basis for the kernel and image of a linear transformation.

At the end of the course, there is a spectacular theorem, The Fundamental Theorem of Linear Algebra. This theorem gives an impressive relationship between the kernel dimension (the number of free variables), the total number of variables, and the number of leading variables. We explain how and why this is so significant and provide additional characterizations of invertible matrices.

Recommended Prerequisites for Subspaces and Their Dimension

I recommend the prerequisite course Linear Transformations. 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.