Linear Algebra with SageMath: A Gentle Introduction

Elementary column operations are the column analogue of elementary row operations. Their precise properties can be deduced by applying elementary row operations to the transpose of a matrix and then transposing the result. In particular, if \(F\) is the elementary matrix corresponding to the elementary column operation \(f\) then \(f(A) = AF\text{.}\) Note that, because transposition changes the order of a product, elementary column operations correspond to post-multiplication by an elementary matrix. Two matrices that are related by a sequence of elementary column operations are said to be column equivalent and two matrices \(A\) and \(B\) are column equivalent if and only if there exists a non singular matrix \(Q\) such that \(B = AQ\text{.}\)