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".

 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