# Row Reduction Over A Field

DevFeed: [Row Reduction Over A Field](<https://devfeed.tech/articles/row-reduction-over-a-field-40251.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2011/12/30/row-reduction-over-a-field/>)

Published: 2011-12-30T15:36:28Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Computing](<https://devfeed.tech/topics/computing.md>), [Optimization](<https://devfeed.tech/topics/optimization.md>)

Tags: [eigenvalues](<https://devfeed.tech/tags/eigenvalues.md>), [eigenvectors](<https://devfeed.tech/tags/eigenvectors.md>), [field](<https://devfeed.tech/tags/field.md>), [linear-algebra](<https://devfeed.tech/tags/linear-algebra.md>), [linear-maps](<https://devfeed.tech/tags/linear-maps.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [matrices](<https://devfeed.tech/tags/matrices.md>), [optimization](<https://devfeed.tech/tags/optimization.md>), [programming](<https://devfeed.tech/tags/programming.md>), [python](<https://devfeed.tech/tags/python.md>), [row-reduction](<https://devfeed.tech/tags/row-reduction.md>)

## AI overview

This tutorial introduces row reduction over a field through matrices representing linear maps between finite-dimensional vector spaces. It explains row equivalence and describes how suitable matrix forms help determine kernels, images, dimensions, eigenvalues, and eigenvectors, with applications to persistent homology and optimization problems.

## Source excerpt

We're quite eager to get to applications of algebraic topology to things like machine learning (in particular, persistent homology). Even though there's a massive amount of theory behind it (and we do plan to cover some of the theory), a lot of the actual computations boil down to working with matrices. Of course, this means we're in the land of linear algebra; for a refresher on the terminology, see our primers on linear algebra.