Row Reduction

Motivation, echelon forms, pivots, rank and what row reduction reveals.

This chapter presents row reduction from motivation and definitions to elementary matrices, rank, pivot structure, what echelon form reveals, geometric interpretation and partial pivoting.

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Chapter contents

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Topic	Pages
Motivation	1
Definition of echelon form	2
Definition of reduced echelon form	3
Utility of row reduction algorithm	3–4
Row reduction algorithm overview	4–5
When are row exchanges needed?	5
Elementary matrices	6–7
Definition & calculation of rank	7–8
Row reduction steps preserve rank	8–9
Row rank = Col rank ⇔ rank(M) = rank(Mᵀ)	8–9
Rows & columns before & after row reduction	10
Reduced row echelon form of a full-rank n × n matrix is the identity matrix	10–11
Proof of linearly dependent columns → non-pivot columns & the corollary	11–13
What echelon form reveals	13–14
What echelon form reveals geometrically	14–15
Partial pivoting	15–16

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