Day 34: Linear Algebra

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Today:

Announcements:
- We have an exam Friday. It will cover everything in chapter III up to (and including) inverses -- but not what we're doing today!
- You have a homework due today.
  1. 1.8, p. 212
  2. 2.17, p. 218
  4. 3.26
Questions about Chapter III?
We are considering "Matrix Operations", in particular
- 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
- 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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