Row Reduction

Echelon form, pivots, rank and structure.

This chapter presents row reduction: how echelon and reduced echelon forms are defined, how pivots reveal rank and pivot columns and why the algorithm is useful across linear algebra.

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Chapter contents

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Topic	Pages
Definition of echelon form	1
Definition of reduced echelon form	1
Utility of row reduction algorithm	2
Row reduction algorithm overview	3
When are row exchanges needed?	4
Elementary matrices	5–6
Vector span	6
Linear independence & linear dependence	7
Definition & calculation of rank	8
Row reduction steps preserve rank	8–9
Row rank = Col rank ⇔ rank(M) = rank(Mᵀ)	9
Rows & columns before & after row reduction	10
Reduced row echelon form of a full-rank n × n matrix is the Identity matrix	10
Proof of linearly dependent columns → non-pivot columns & the corollary	11–12
What echelon form reveals	13
Partial pivoting	14–15

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