10/01 - Vectors & Functions: Difference between revisions

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*What does the <math> \hat b_m </math> represent, say compared to <math> \hat b_j</math>?
*What does the <math> \hat b_m </math> represent, say compared to <math> \hat b_j</math>?
**<math> \hat b_m </math> is a unit vector in the direction we're interested in finding the new coefficient for the new basis set -- rewrite this
**<math> \hat b_m </math> is a unit vector in the direction we're interested in finding the new coefficient for the new basis set -- rewrite this
**<math> \hat b_j </math> is
**<math> \hat b_j </math> is a unit vector for one direction in our new basis set
*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?
*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 <math>k_m \,\!</math> instead of <math>k_j\,\!</math>?
*Why did you decide to make it <math>k_m \,\!</math> instead of <math>k_j\,\!</math>?

Revision as of 16:11, 10 November 2008

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:

Thus is the length of squared

Questions

  • What does the represent, say compared to ?
    • is a unit vector in the direction we're interested in finding the new coefficient for the new basis set -- rewrite this
    • is a unit vector for one direction in our new basis set
  • 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 instead of ?
    • Completely arbitrary, the end result is the same either way
  • is a unit vector and is a vector