- First off, we have an exam Friday. It will cover everything in
chapter III up to (and including) inverses -- but not what we're doing today!
- How did last class go? Did you understand my car-buying-experience example, and how
to carry out your second project?
Many phenomena in nature are linear. For example, Broad or Dainty, Crocodilian Snouts Pack Same Chomp (New York Times, ). Here's the essential graphic.
To re-phrase the essential question to be answered for your project, one might ask this:
"Which crocodilian species packs the most fearsome bite, pound for pound?" - You have a homework due today.
- Homework returned:
- 2.15, p. 208. "No." does not cut it. You need to explain.
- 2.18. And referring to the corollary that the author referred to is not acceptible. I know that's what the author said -- but you are not the author. Your answers should generally look entirely different from the author's.
- 2.19: one intuitive way to think about these is to ask
"What is the image of the unit circle under the map?"
- 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
then M is a change of basis matrix.
- Change of basis matrix:
- Theorems:
- 1.4 Lemma: A matrix changes bases if and only if it is nonsingular.
- Examples:
- 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.
- 1.6, p. 239
- 1.7