On this page, we collect resources about Linear Transformations defined by matrices.


The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to “read ahead” before each class - but can equally be used in a more traditional course setting.

Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch - you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.

Basic learning objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

Advanced learning objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

To prepare for class

  • Watch this video by 3Blue1Brown which explains Linear Transformations defined by \(2\times 2\) matrices:

  • Experiment with these interactive applets visualizing \(2\times 2\) linear transformations:

    • Pick any image you like from the internet (e.g. use Google Images Search, then copy and paste the image URL) and transform it:

      https://web.ma.utexas.edu/users/ysulyma/matrix/ (written by Yuri Sulyma)

      (Note: press “enter” after making any change to update the image)

    • Pick image vectors for the two basis vectors \(\vec{i}\) and \(\vec{j}\) and then see how the plane gets transformed to achieve this movement (in the style of the 3Blue1Brown video):

      https://shadanan.github.io/MatVis/ (written by Shad Sharma)

    • You can also use Geogebra to create your own applets, see for example this website (especially near the bottom of the page) for a tutorial on how to do that:

      http://www.malinc.se/math/linalg/vectorsen.php (written by Malin Christersson)

  • Watch this video by 3Blue1Brown which explains how Matrix multiplication can (and should) be interpreted as the Composition of Linear Transformations:

  • Watch this video by 3Blue1Brown which shows Linear Transformations defined by \(3\times 3\) matrices:

  • Watch this video by 3Blue1Brown which illustrates Linear Transformations defined by non-square matrices:

After class

  • Watch the first 7 minutes of this video by 3Blue1Brown which gives a geometric interpretation of the inverse of a matrix, and its relationship to solving systems of equations:


Gabriel Indurskis Avatar Gabriel Indurskis






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