Dave4Math / Courses / Linear Transformations

Linear Transformations

Master linear transformations and their inverses. Learn about linear transformations used in geometry, matrix products, and inverse matrices. With this course, finding the inverse of a matrix is no longer challenging.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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Are you a participant in your math class? If not, is it because you are not confident, or is it because you don’t have the opportunity to engage with your teacher. Introducing Dave’s online math course: Linear Transformations. Dave’s lessons are engaging and compel you to discuss mathematics with other students.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in linear transformations.
  • 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 transformations and their inverses, linear transformations in geometry, and matrix products.


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 Transformations

  • Introduction to Linear Transformations
  • Definition of Linear Transformation
  • Characterization of Linear Transformation
  • Scalings
  • Orthogonal Projections
  • Reflections
  • Orthogonal Projections and Reflections in Space
  • Rotations
  • Rotations Combined with a Scaling
  • Matrix Multiplication
  • Matrix Algebra
  • Block Matrices
  • Powers of Transition Matrices
  • Invertible Functions
  • Invertible Matrices
  • A Criterion for Invertibility
  • The Inverse of a Block Matrix
  • Definition and Examples of Linear Maps
  • Linear Maps and Basis of Domain
  • Algebraic Operations on L(V,W)
  • Algebraic Properties of Products of Linear Maps
  • Null Space and Injectivity
  • Range and Surjectivity
  • Fundamental Theorem of Linear Maps
  • Homogeneous System of Linear Equations
  • Inhomogeneous System of Linear Equations
  • Representing a Linear Map by a Matrix
  • Matrices
  • The Matrix of the Sum of Linear Maps
  • The Matrix of a Scalar Times a Linear Map
  • Matrix Multiplication
  • The Matrix of the Product of Linear Maps
  • Linear Combination of Columns
  • Invertible Linear Maps
  • Isomorphic Vector Spaces
  • Linear Maps Thought of as Matrix Multiplication
  • Operators
  • Products of Vector Spaces
  • Products and Direct Sums
  • Quotients of Vector Spaces
  • The Dual Space and the Dual Map
  • The Null Space and Range of the Dual of a Linear Map
  • The Matrix of the Dual of a Linear Map
  • The Rank of a Matrix

Course Description

We begin this course with a thorough explanation of what a (real) vector space is. We provide proof of several elementary properties of vector spaces illustrating a vector space’s axioms and how to use them.

Next, we discuss what a linear transformation is by using a matrix definition and using the standard vectors of real vector spaces. Then, we work through several examples trying to understand if a given transformation is a linear transformation. After that, we characterize when a linear transformation has the properties of linearity.

In the next lesson, we study linear transformations on the Cartesian plane that are commonly used in plane geometry. In particular, we learn scalings, orthogonal projections, reflections, rotations, and shears. With each type of transformation, we motivate, prove, and then illustrate with several examples.

Next, we study what an invertible transformation is. We begin by understanding that an invertible transformation (if it exists) is an invertible function of a linear transformation (as a function). Several examples are demonstrated by applying elementary row operations. After that, a characterization of when a square matrix is invertible is given in terms of 1) reduced row-echelon form, 2) the rank of the corresponding system, and 3) invertibility of the matrix. Finally, we relate back to when a matrix is invertible with linear systems, as discussed in the previous course.

Recommended Prerequisites for Linear Transformations

I recommend the prerequisite course Linear Equations. 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.