Day 8: Linear Algebra

Last Time

Next Time

Today:

Announcements:
- You have an assignment due today.
- There will be a test next Friday, over the material of Chapter One.
We'll begin by finishing Section One.II.2: Linear systems. Linear Geometry of n-Space.Length and Angle Measures
- Definitions of this section
  - rely on the dot product:
    
    ${\overline{u}\cdot\overline{v}=u_1v_1+u_2v_2+\ldots+u_nv_n}$
  - length of a vector:
    
    ${||\overline{u}||=\sqrt{\overline{u}\cdot\overline{u}}}$
  - angle between two vectors
    
    ${\overline{u}\cdot\overline{v}=||{\overline{u}}||||{\overline{v}}||\cos(\theta)}$
  - orthogonal (perpendicular) vectors ${\overline{u}\cdot\overline{v}=0}$
- Theorems:
  - Triangle Inequality
  - Cauchy-Schwartz Inequality (corollary)
  We are going to prove this result, following the author's proof (throwing in a few lemmas along the way).
- Examples:
  - 2.10c
  - 2.11b
  - 2.13
  - 2.24
  - 2.27
Now, on to Section One.III.1: Linear systems.Reduced Echelon Form.Gauss-Jordan Reduction
- 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.
- 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.
  - GJ 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 Gauss-Jordan form.
    - But it requires more (and often unnecessary) arithmetic -- just backsolve ASAP....
  - row equivalent matrices -- interchangeable by elementary row operations
  - row equivalence classes.
- Theorems (Lemmas only in this section -- they're moving us toward proving a theorem -- 2.6 Theorem, p. 57 -- in the last section of Chapter One -- that 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)
- Examples:
  - 1.7c, p. 51
  - 1.8c
  - 1.10
  - 1.11b

Website maintained by Andy Long. Comments appreciated.