CS223 · Chapter 1 · Annotated SlidesRow Reduction and Echelon Forms
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Row Reduction and Echelon Forms

Echelon and reduced echelon form, pivot positions and columns, the row reduction algorithm, basic and free variables, parametric solutions, and existence and uniqueness of solutions.

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Every original slide is shown unchanged. Yellow marks on a slide point to a numbered note: hover or tap a mark to jump to its note, and hover a note to see its marks. Questions tagged Past exam appeared on a real paper; Prediction questions are likely angles, not confirmed. Pick an answer to each question: a wrong pick tells you why it is wrong, so you can try again.

All 20 slides
  1. 1 Row Reduction and Echelon Forms
  2. 2 Row Reduction and Echelon Forms
  3. 3 Echelon Form
  4. 4 pivots
  5. 5 Algorithms to produces a matrix in Echelon Form
  6. 6 Example
  7. 7 Algorithms to produce a matrix in Reduced Echelon Form
  8. 8 Apply elementary row operations to transform
  9. 9 Solutions of Linear Systems
  10. 10 Example
  11. 11 Parametric representation
  12. 12 Note
  13. 13 Example
  14. 14 Slide 14
  15. 15 Slide 15
  16. 16 Exercise
  17. 17 Exercise
  18. 18 Exercise
  19. 19 Exercise
  20. 20 Exercise
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Row Reduction and Echelon Forms

Slide 1: Row Reduction and Echelon Forms
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Row Reduction and Echelon Forms

Slide 2: Row Reduction and Echelon Forms
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Echelon Form

Slide 3: Echelon Form

Notes

  1. Echelon form: zero rows at the bottom, each leading entry to the right of the one above, zeros below each leading entry. REDUCED: leading entries are 1 and are the only nonzero entries in their columns.

  2. Trap A matrix has many echelon forms but only ONE reduced echelon form.

Exam angle

PredictionQ1

Each matrix is row equivalent to exactly one reduced echelon matrix.

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pivots

Slide 4: pivots

Notes

  1. Pivot position: location of a leading 1 in the reduced echelon form; pivot column: a column containing one.

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Algorithms to produces a matrix in Echelon Form

Slide 5: Algorithms to produces a matrix in Echelon Form
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Example

Slide 6: Example
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Algorithms to produce a matrix in Reduced Echelon Form

Slide 7: Algorithms to produce a matrix in Reduced Echelon Form
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Apply elementary row operations to transform

Slide 8: Apply elementary row operations to transform
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Solutions of Linear Systems

Slide 9: Solutions of Linear Systems
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Example

Slide 10: Example
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Parametric representation

Slide 11: Parametric representation

Notes

  1. Write the general solution in parametric form: basic variables in terms of the free ones.

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Note

Slide 12: Note
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Example

Slide 13: Example

Notes

  1. Existence and uniqueness: consistent ⇔ the last column of the augmented matrix is NOT a pivot column (no row [0 … 0 | b], b ≠ 0). If consistent: unique ⇔ no free variables; infinitely many ⇔ at least one free variable.

Exam angle

Past exam · Major 1 (251)

Related question: this topic is examined

PredictionQ1

An augmented matrix reduces to a row [0 0 0 | 5]. The system is:

PredictionQ2

A consistent system with 3 equations, 5 unknowns has:

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Slide 14

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Slide 15

Slide 15: Slide 15
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Exercise

Slide 16: Exercise
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Exercise

Slide 17: Exercise
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Exercise

Slide 18: Exercise
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Exercise

Slide 19: Exercise
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Exercise

Slide 20: Exercise