Uploaded April 2022 | Updated September 2026, 2 weeks ago

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In this video tutorial I discuss the nullspace of a matrix. I use the sympy package in python do explain the nullspace. Python and the symbolic python package sympy makes it easy to work with mathematical symbols. It is excellent at linear algebra.

In the video I explain what the nullspace is and go through examples that include all the types of matrices that you will come across. I also discuss linear independence and make clear how the nullspace differs from the column space. At the end of the tutorial I also show you how to prove that the nullspace is a vector subspace.

00:00 Introduction
05:59 Square matrix with linearly independent column vectors
11:03 Square matrix with dependent column vector
11:16 Rectangular matrix with more columns than rows with a single dependent column vector
20:01 Rectangular matrix with more columns than rows and more than one dependent column vector
21:48 Rectangular matrix with more rows than column and with independent column vectors
24:39 Rectangular matrix with more rows than columns and a dependent column vector
27:18 Proof that the nullspace is a subspace