Homework One

We know that v = \tilde v.

We know that some matrix operating on v gives us \tilde v.

We know that some other matrix operating on \tilde v gives us v.

  • Av = \tilde v
  • B \tilde v= v

Let’s do some math to make the A disappear… A A^{-1}=I

  • A^{-1} A v = A^{-1} \tilde v
  • v = A^{-1} \tilde v
  • B = A^{-1}

This doesn’t tell us what A is, but it tells us that if we know what A is and we can get the inverse of A, we know B.

The above procedure focused on A. We can do a similar procedure that focuses on B:

  • A v= \tilde v
  • B \tilde v = v
  • B^{-1} B \tilde v = B^{-1} v
  • \tilde v = B^{-1} v

We can now generalize what we said above to the follow:

Knowledge of one gives us the other if we can calculate the inverse of the one.

It also forces Mother Nature to cough up one more fact: a matrix like B or A must be such that an Inverse Matrix exists.