Day 36: Linear Algebra

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Today:

Announcements:
- Your exams returned. Ouch!
  - I still have three to grade (took them late), but preliminary results indicate that there is a lot of abstract nonsense going on.
    Here's the deal: you have until Friday to make corrections on a fresh copy of the exam. Then submit both the original and the corrected copy to me on Friday, for 50% of the points corrected. So, if you'll fix everything, then you'll move halfway to the goal (of 100%, of course).
  I need the exams back, to assist in grading the other three.
- Projects are returned.
  Some of them were too close to the author's solutions. If you didn't distinguish your solutions from the author's solutions, of what value was this project? I could have cut and pasted the author's solutions into a word document.... but that's not worth anything. I don't want to measure your cutting and pasting skills.
- If you're working on project 2 with a partner:
  - Change in the rules: Each should write up the problem separately. Don't give me a joint report.
  - Each of you should "do the whole problem": the point of having a partner is not to cut the work in half, but to have an extra head.
- You have a new homework assignment, due Monday next.
- Tomorrow we begin Chapter Four: Determinants
We are considering "Matrix Operations", in particular
- Three.V.1: Maps Between Spaces.Change of Basis.Changing Representations of Vectors
  1. Definitions:
    - Change of basis matrix:
    - 1.2 Lemma (this really defines a change of basis):
      Left-multiplication by the change of basis matrix for B, D converts a representation with respect to B to one with respect to D.
      Conversely, if left-multiplication by a matrix changes bases ${M\cdot{Rep_B(\overline{v})}={Rep_D(\overline{v})}$ then M is a change of basis matrix.
  2. Theorems:
    1. 1.4 Lemma: A matrix changes bases if and only if it is nonsingular.
  3. Examples:
    1. Problem 2.18, p. 208, is what it's all about: if we change bases so that a matrix is represented by a diagonal matrix, then the homomorphism is easy to apply.
    2. 1.6, p. 239
    3. 1.7
- Three.V.2: Maps Between Spaces.Change of Basis.Changing Map Representations
  1. Definitions:
    - matrix equivalent: Same-sized matrices ${H}$ and ${\hat{H}}$ are matrix equivalent if there are nonsingular matrices P and Q such that ${H=P\hat{H}Q}$ .
  2. Theorems:
    1. Corollary: Matrix equivalent matrices represent the same map, with respect to appropriate pairs of bases.
    2. Theorem 2.6:
    3. Corollary Two same-sized matrices are matrix equivalent if and only if they have the same rank. That is, the matrix equivalence classes are characterized by rank.
  3. Examples:
    1. "2.2 Example One reason to change bases is that the matrix may be simpler." p. 242
    2. 2.10, p. 246
    3. 2.12

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