Linear algebraLesson 3 of 5 in the course Systems of linear equations

Row Echelon Forms: RREF, Pivots, Rank, and Integer Variants

5 flashcards · 5 quiz questions · 5 match pairs · 4 fill-in sentences · English

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Summary

This lesson introduces row echelon form, the staircase pattern that appears after Gaussian elimination. Recognizing that pattern is an important first step in solving linear systems and analyzing matrices.

The leading entry is also called a pivot. The word echelon refers to a level or step of a ladder, which is a helpful way to picture the arrangement of pivots.

All flashcards

What defines reduced row echelon form (RREF)?
Row echelon form with leading ones and zeros elsewhere in each pivot column.
What is a pivot column?
A column containing the leading entry of a nonzero row.
What are the elementary row operations?
Swapping rows, scaling a row by a nonzero scalar, and adding a multiple of one row to another.
What is a free variable in a linear system?
A variable whose column does not contain a pivot.
What is Hermite normal form useful for?
An echelon-style form for integer matrices that can avoid introducing denominators.

Quiz questions

  1. Which is an elementary row operation that preserves the solution set of a linear system when applied to its augmented matrix?Answer: Replace a row by itself plus a scalar multiple of another row.
  2. What does Gauss–Jordan elimination do beyond producing row echelon form?Answer: It eliminates entries above pivots as well as below them.
  3. What row in an augmented matrix signals an inconsistent system?Answer: A row with zero coefficients and a nonzero constant.
  4. Which form is unique for a given matrix after valid elementary row operations?Answer: Reduced row echelon form.
  5. Why can Hermite normal form be useful when working with integer-coefficient matrices?Answer: It can provide an echelon-style form without introducing denominators.

Based on the Wikipedia article Row echelon form, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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