On the Column and Row Ranks of a Matrix

TL;DR

This paper provides a self-contained introduction to the concepts and tools related to the rank of a matrix, emphasizing its fundamental role in linear algebra and applications.

Contribution

It offers an accessible overview of matrix rank concepts and mathematical tools, facilitating understanding for applied linear algebra without covering all advanced results.

Findings

Clarifies the fundamental theorem relating matrix rank, rows, and columns.

Introduces key concepts and tools for understanding matrix rank.

Serves as an educational resource for applied linear algebra.

Abstract

Every m by n matrix A with rank r has exactly r independent rows and r independent columns. The fact has become the most fundamental theorem in linear algebra such that we may favor it in an unconscious way. The sole aim of this paper is to give a self-contained introduction to concepts and mathematical tools for the rank of a matrix in order to seamlessly introduce how it works in applied linear algebra. However, we clearly realize our inability to cover all the useful and interesting results concerning this topic and given the paucity of scope to present this discussion, e.g., a proof via the injective linear map. We refer the reader to literature in the field of linear algebra for a more detailed introduction to the related fields.