Echelon Matrix Forms

 

Linear Algebra 2: Echelon Matrix Forms

Welcome back to the second essay of my ongoing series on the basics of Linear Algebra, the foundational math behind machine learning. In my previous article, I introduced linear equations and systems, matrix notation, and row reduction operations. This article will walk through the echelon matrix forms: row echelon form and row reduced echelon form and how both can be used to solve linear systems. This article would best serve readers if read in accompaniment with Linear Algebra and Its Applications by David C. Lay, Steven R. Lay, and Judi J. McDonald. Consider this series as an external companion resource.

The above operations can be applied to a matrix to transform that matrix into its row echelon form. A given m x n matrix, where m is the number of rows and n is the number of columns is said to be in row echelon form when:

Any rows where all entries are zero are below rows where at least one entry is non-zero.

All leading entries of a row (first entry from the left that is non-zero) are in a column to the right of the row above it.

All entries in a column below a leading entry are zero.

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