On this page, we discuss linear transformations between general vector spaces. When the spaces are finite-dimensional, we also study their matrix representations with respect to various choices of bases, and how to change bases.


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.

  • Find the matrix representation of the “taking the derivative” linear transformation, in the space of real polynomials of degree \(4\) or less.

Advanced learning objectives

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

  • Find the matrix representation of any linear transformation between any two vector spaces, by studying what happens to the input basis vectors.
  • Use a basis change matrix to convert from one basis to another in \(\mathbb{R}^n\).

To prepare for class

  • Watch the second half of this video (again) by 3Blue1Brown, starting at 6:20, which shows how to express the linear transformation “taking the derivative” in the space of polynomials as a matrix:

  • Watch this video by Lorenzo Sadun about finding the transformation matrix for a transformation between general vector spaces, on the example of polynomials:

After class

  • Watch this video by 3Blue1Brown about how to change from one basis of \(\mathbb{R}^n\) to another, using a basis change matrix:


Gabriel Indurskis Avatar Gabriel Indurskis






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