Day 37: Linear Algebra

• Reminder: your exams should be corrected and returned by this Friday, at classtime. Here is a link to the exam, so that you can work on a fresh copy.
• Second Reminder: if you're working on Project 2 with a partner then there's been one change in the rules of the game:
  • 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.

Three.V.2:

1. Definitions:
   • matrix equivalent: Same-sized matrices and are matrix equivalent if there are nonsingular matrices P and Q such that .

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

4. Upshot: if we make the right choice of bases, a homomorphism is represented by that lovely diagonal matrix above, with just ones and zeros on its diagonal (how many of each depends on its rank). Talk about simpler!

• I want to focus our understanding of Determinants on three things:
  1. The job(s) determinants do
  2. One method of calculating determinants
  3. The geometric interpretation of the determinant.

  Now for some details:

  1. The job determinants do:

     First and foremost, determinants tell us if a transformation is invertible or not.

     Consider a homomorphism h associated with matrix H. Then if the determinant of H is 0, then the matrix is singular (and hence non-invertible).

     In a bit you'll see that they also tell us something about a volume of interest....

  2. One method of calculating determinants:

     For this, we consider a few cases, and then arrive at a recursive formula for their calculation:

     1. One-dimensional case
     2. Two-dimensional case
     3. Three-dimensional case
     4. Generalization (see "Other Formulas", p. 324)

  3. The geometric interpretation of the determinant.
     • Recall that homomorphisms tranform circles into ellipses, spheres into ellipsoids, hyperspheres into hyperellipsoids, etc. Whatever the ball is in n dimensions, it gets squished into an ellipsoidal ball in n dimensions.

     • So if we start with a unit ball, one question of interest is what happens to its volume?

     • The determinant of matrix A tell how the volume of the image of the unit sphere scales under the associated homomorphism a: it scales by a factor