# 10/01 - Vectors & Functions

From Class Wiki

## Contents

## 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 , is it always the projection of onto and not the opposite way around?
- Then is the picture assuming something is a unit vector?
- You will have to choose whether you're interested in projecting A onto B or B onto A. The lengths will be the same, but the direction will be different.

- 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