Day 9: Linear Algebra

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Today:

Announcements:
- You have an assignment due today.
- There will be a test this Friday, over the material of Chapter One -- which we finish up today.
Last time we finished Section One.II.2: Linear systems. Linear Geometry of n-Space.Length and Angle Measures
Now, on to Section One.III.1 and III.2: Linear systems.Reduced Echelon Form.Gauss-Jordan Reduction and Linear systems.Reduced Echelon Form.Row Equivalence
- Summary and major points:
  - The process of Gauss reduction is not-unique. However we have seen that it preserves the solution set of a linear system. Hence, every reduced augmented matrix system represents an equivalent system, from the standpoint of the solutions.
  - This section shows us how to reduce to a single elementary form (Gauss-Jordan form) that exposes the solution in a nice way.
  - Motivation: actually the first time Zach got up to solve a system, he reduced it all the way to Gauss-Jordan form -- to a diagonal matrix, containing just ones on the diagonal, and we noticed then that we could just read off the solution that way. That's what we're doing in this section -- just pushing on with our row-reductions until we get it in "Zach-form".
- Definitions of this section:
  - Gauss-Jordan reduction
    the matrix has only ones for leading entries in rows, and each leading one is alone in its column.
    In a way, it "represents" its column.
  - Gauss-Jordan reduction leads to an augmented system in reduced echelon form:
    - The solution can be just read off...
    - We can also see the dimension of the solution easily (back to Jenny's question regarding 2.29, p. 19) -- the example I gave was an example in Gauss-Jordan form.
    - But it requires more (and often unnecessary) arithmetic -- sometimes you just want to backsolve ASAP, and not sit around admiring your solution's "exposure"....
  - row equivalent matrices -- interchangeable by elementary row operations
  - row equivalence classes.
- Theorems (Lemmas only in the first section, and primarily lemmas in the second -- they're moving us toward proving a theorem -- 2.6 Theorem, p. 57: each matrix is row equivalent to a unique reduced echelon form matrix)
  - Elementary row operations are reversible.
  - Between matrices, "reduces to" is an equivalence relation.
    An equivalence relation is one such that
    1. A is equivalent to itself (reflective)
    2. A is equivalent to B iff B is equivalent to A (symmetric)
    3. If A is equivalent to B and B is equivalent to C, then A is equivalent to C (transitive)
  - (Linear Combination Lemma) A linear combination of linear combinations is a linear combination.
  - Where one matrix reduces to another, each row of the second is a linear combination of the rows of the first.
  - In an echelon form matrix, no nonzero row is a linear combination of the other rows.
  - If two echelon form matrices are row equivalent then the leading entries in their first rows lie in the same column. The same is true of all the nonzero rows -- the leading entries in their second rows lie in the same column, etc.
  - Here's the big theorem: Each matrix is row equivalent to a unique reduced echelon form matrix.
- Examples that illustrate the concepts in this section:
  - 1.7c, p. 51
  - 1.8c
  - 1.10
  - 1.11b
  - 2.10cd, p. 59
  - 2.12a
  - 2.16
  - 2.18b
  - 2.23

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