Matrix transformations

This course is a part of Linear algebra, a 3-course Topic series from Khan Academy.

Understanding how we can map one set of vectors to another set. Matrices used to define linear transformations.

This course contains 7 segments:

Functions and linear transformations

People have been telling you forever that linear algebra and matrices are useful for modeling, simulations and computer graphics, but it has been a little non-obvious. This tutorial will start to draw the lines by re-introducing you functions (a bit more rigor than you may remember from high school) and linear functions/transformations in particular.

Linear transformation examples

In this tutorial, we do several examples of actually constructing transformation matrices. Very useful if you've got some actual transforming to do (especially scaling, rotating and projecting) ;)

Transformations and matrix multiplication

You probably remember how to multiply matrices from high school, but didn't know why or what it represented. This tutorial will address this. You'll see that multiplying two matrices can be view as the composition of linear transformations.

Inverse functions and transformations

You can use a transformation/function to map from one set to another, but can you invert it? In other words, is there a function/transformation that given the output of the original mapping, can output the original input (this is much clearer with diagrams). This tutorial addresses this question in a linear algebra context. Since matrices can represent linear transformations, we're going to spend a lot of time thinking about matrices that represent the inverse transformation.

Finding inverses and determinants

We've talked a lot about inverse transformations abstractly in the last tutorial. Now, we're ready to actually compute inverses. We start from "documenting" the row operations to get a matrix into reduced row echelon form and use this to come up with the formula for the inverse of a 2x2 matrix. After this we define a determinant for 2x2, 3x3 and nxn matrices.

More determinant depth

In the last tutorial on matrix inverses, we first defined what a determinant is and gave several examples of computing them. In this tutorial we go deeper. We will explore what happens to the determinant under several circumstances and conceptualize it in several ways.

Transpose of a matrix

We now explore what happens when you switch the rows and columns of a matrix!