LA38 Why is Column Rank = Row Rank?

For #matrices , the max number of linearly independent rows (i.e., row rank) equals the max number of linearly independent columns (i.e., column rank) and is called the #rankofamatrix. But why are these two numbers always the same? We give a proof.#linearalgebra Subscribe ‪@Shahriari‬ for more undergraduate math videos.
00:00 Introduction
00:17 Statement of the Problem: row rank = column rank
01:11 Example of row rank and column rank
02:35 Outline of the Proof
05:06 Review: Row & column space, row & column rank
08:35 Review: Elementary Row Operations and Row & Column space
12:05 Basis for row & column spaces of rref(A)
15:03 Proof that row rank = column rank for all matrices

This is a video in a series of lectures on linear algebra. The series is a rigorous treatment meant for students with no prior exposure to linear algebra. In this full undergraduate course in linear algebra, general vector spaces and linear transformations are emphasized.
This is also a lecture in a series of lectures on Matrices and their properties. Matrices play an important role in Linear Algebra. We use them in the study of vector spaces and, in return, the theory of vector spaces is used to understand matrices.

For an annotated list of available Linear Algebra videos see
https://pomona.box.com/s/oahafa797tj5...
Linear Algebra YouTube Playlist:   • Linear Algebra, a Retro introduction  

Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA, U.S.A.