10/01 - Vectors & Functions

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Vectors & Functions

  • How to related the vector v to the sampling?

We could sample a continuous function every T seconds, creating a "bar graph".

  • Where is a rectangle 1 unit high and T units wide

In an effort to make this more exact, will will continue to shrink the rectangle down to the Dirac Delta function,

By using the Dirac Delta function the summation becomes an integral

Changing from one orthogonal basis set to another

We have a vector and wish to change it to . We know each basis set, and their relationship to each other. We are trying to find the coefficients, (the ) that go with the new basis set.

  • Working from the basis set:
  • Working from the basis set:
  • Now taking the that was derived from both basis sets and equating them:

Defining

Taking from the previous section:

Questions

  • How did you get the last two lines of the last page?
  • What does the represent, say compared to ?
  • When you do the dot product of say A \cdot B, is it always the projection of A onto B and not the opposite way around?
  • Why did you decide to make it k_m instead of k_j?