Overview
On this page, we discuss two special subspaces of \(\mathbb{R}^n\) associated to a matrix, the nullspace (or kernel) and the column space (or image or range) - as well as their dimensions, the nullity and rank.
Important
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 nullspace of a matrix and a basis for it.
- Find the nullity of a matrix (i.e. the dimension of the nullspace).
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 a basis for the column space of a matrix.
- Use the Rank Theorem to quickly determine the rank or nullity when the other value is given, and to give maximal or minimal possible values for each.
To prepare for class
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Watch this video by 3Blue1Brown (you probably have already watched the first 7 minutes previously) which introduces the concepts of rank and column space from a geometric point-of-view:
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Watch this video by Khan Academy which shows an example calculation of the nullspace of a \(3\times 4\) matrix:
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Watch this video by Joy Zhou which shows how to find a basis for the column space (or range) of a matrix:
After class
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Watch this video by TheTrevTutor which introduces the row space of a matrix, and explains the surprising and important Rank Theorem, which states that the rank and nullity of an \((m\times n)\)-matrix always add up to \(n\) (the dimension of the target space):
