# 10/01 - Vectors & Functions

## Vectors & Functions

• How to related the vector v to the sampling?

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

$f(t)= \sum_{i=0}^{N-1} \underbrace{f(i T)}_{coefficients} \cdot \underbrace{p(t - i T)}_{basis functions}$

• Where $p(t) \,\!$ 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, $\delta \,\!$

• $\delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}$
• $\int_{-\infty}^\infty \delta(x) \, dx = 1.$

By using the Dirac Delta function the summation becomes an integral

$f(t) = \int_{-\infty}^{\infty} f(u) \cdot \delta (t - u)\, du$

## Changing from one orthogonal basis set to another

We have a vector $\hat v = \sum_{j=1}^3 a_j \hat a_j$ and wish to change it to $\hat v = \sum_{j=1}^3 b_j \hat b_j$. We know each basis set, and their relationship to each other. We are trying to find the coefficients, (the $b_j \,\!$) that go with the new basis set.

• Working from the $\hat a$ basis set:
$\hat v \cdot \hat b_m= \sum_{j=1}^3 v_j \hat a_j \cdot \hat b_m = \sum_{j=1}^3 v_j \underbrace{\left (\hat a_j \cdot \hat b_m \right )}_{proj of \hat a_j on \hat b_m}$
• Working from the $\hat b$ basis set:
$\hat v \cdot \hat b_m= \sum_{j=1}^3 b_j \hat b_j \cdot \hat b_m = \sum_{j=1}^3 b_j \underbrace{\left (\hat b_j \cdot \hat b_m \right )}_{proj of \hat b_j on \hat b_m}= \sum_{j=1}^3 b_j k_m \delta mj = k_m \sum_{j=1}^3 b_j \delta mj= b_m k_m \sum_{j=1}^3 = b_m k_m$
• Now taking the $\hat v \cdot \hat b_m$ that was derived from both basis sets and equating them:
$b_m k_m = \sum_{j=1}^3 v_j \hat a_j \cdot \hat b_m \Longrightarrow b_m = \frac{1}{k_m} \sum_{j=1}^3 v_j \left (\hat a_j \cdot \hat b_m \right )$

## Defining $k_m \,\!$

Taking $k_m \,\!$ from the previous section:

$\hat b_j \cdot \hat b_m = k_m \delta mj \Longrightarrow \hat b_m \cdot \hat b_m = k_m \Longrightarrow \left | \hat b_m \right |^2 = k_m$

Thus $k_m \,\!$ is the length of $\hat b_m$ squared

## Questions

• What does the $\hat b_m$ represent, say compared to $\hat b_j$?
• $\hat b_m$ is a unit vector in the direction we're interested in finding the new coefficient for the new basis set -- rewrite this
• $\hat b_j$ is a unit vector for one direction in our new basis set
• When you do the dot product of say $\vec A \cdot \vec B$, is it always the projection of $\vec A$ onto $\vec B$ and not the opposite way around?
• $\vec A \cdot \vec B = \left | \vec A \right | \left | \vec B \right | \cos \theta$
• 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 $k_m \,\!$ instead of $k_j\,\!$?
• Completely arbitrary, the end result is the same either way
• $\hat x$ is a unit vector and $\vec x$ is a vector