Day 39: Linear Algebra

Notice that they use only mileage, not year. While mileage may be a good indicator of age, it is not perfect. The van that I purchased had very few miles for its age, and I believe that that showed (and has made it a good vehicle for us).

Rust (and other kinds of damage, e.g. degradation of rubber parts) may be more reliably indicated by age than by mileage.

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)

   Details:

   • Definition 1.2, p. 325: minor and cofactor
     

   • Theorem 1.5, p. 325:
     

     So we find the determinant of an matrix by computing a sum of n determinants of smaller matrices, each of size

   • Here's my code in lsp for doing the recursive calculation of the determinant for any dimension matrix.

     Let's check some of these by hand....

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.

     We'll check this argument, following the article above. One thing that we'll want to check is that the equation given in (1) is actually an ellipse.

     There is vocabulary here on page 204 of the article that we have not encountered yet, but which is the focus of the rest of the course: what is an eigenvalue, and what is an eigenvector?

   • 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

   • That being the case, what happens to volume if we consider a homomorphism associated with a singular matrix?

It starts off with complex numbers, so if you're already suitably familiar with those, you can even cut out some of that reading!

What happens when we multiply the two vectors by H?

These vectors have a special name: eigenvectors. And the scalars by which they're scalled are called eigenvalues.

Are all matrices "diagonalizable"? (Can you think of a homomorphism from we've studied geometrically where no vector will have as an image a multiple of itself?

We can re-write that as

obviously (since I is the identity matrix); now we take everything over to the lefthand side,

i.e.