Uploaded April 2022 | Updated September 2026, 2 weeks ago

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In this video tutorial I discuss the column space of a matrix. I use the sympy package in python to explain the column space. 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 continue on from the previous video tutorial on the nullspace of a matrix. I explain what the column space is and go through examples that include all the types of matrices that you will come across. I also discuss what effect elementary row operations have on the column space. At the end of the tutorial I also show you how to prove that the column space is a vector subspace.

00:00 Introduction
03:55 Review of the nullspace
05:56 Square matrix with linear independence
09:19 Square matrix with linear dependence
17:01 Rectangular matrix with more columns than rows
20:43 Rectangular matrix with more rows than columns and linear independence
22:27 Rectangular matrix with more rows than columns and linear dependence
24: 16 Proof that the column space is a vector subspace