CS223 · Annotated SlidesChapter 1 — Row Reduction and Echelon Forms
Chapter 1 · Annotated slides

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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Slide 1: Row Reduction and Echelon Forms
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    Slide 2: Row Reduction and Echelon Forms
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      Slide 3: Slide 3
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      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.
      Exam trapA matrix has many echelon forms but only ONE reduced echelon form.
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      True / False

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

      Slide 4: pivots
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      1. Pivot position: location of a leading 1 in the reduced echelon form; pivot column: a column containing one.
      Slide 5: Algorithms to produces a matrix in Echelon Form
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        Slide 6: Example
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          Slide 7: Algorithms to produce a matrix in Reduced Echelon Form
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            Slide 8: Apply elementary row operations to transform
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              Slide 9: Solutions of Linear Systems
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              1. Basic variables correspond to pivot columns; free variables to the others.
              Slide 10: Example
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                Slide 11: Parametric representation
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                1. Write the general solution in parametric form: basic variables in terms of the free ones.
                Slide 12: Note
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                  Slide 13: Example
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                  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.

                  Related real-exam questions

                  Major 1 (251) · this topic is examined
                  🎯 How this can come up · 2 questions
                  MCQ

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

                  MCQ

                  A consistent system with 3 equations, 5 unknowns has:

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                      Slide 16: Exercise
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                        Slide 17: Exercise
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                          Slide 18: Exercise
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                            Slide 19: Exercise
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                              Exercise

                              Find the solution of the following system

                              x + 2 ݕ − ݖ = 3 2x − ݕ + 3ݖ = 5

                              3x + 6ݕ − 3ݖ = 10

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