dot product practice - linear algebra 101

Info Theory homework

from numpy import *
from matplotlib.pyplot import plot
import matplotlib.pyplot as plt
import numpy as np

Here is one set of basis vectors.

$$ \hat{a}_1 = [1, 0, 0, 0] \\ \hat{a}_2 = [0, 1, 0, 0] \\ \hat{a}_3 = [0, 0, 1, 0] \\ \hat{a}_4 = [0, 0, 0, 1] \\ $$

And here is another set of basis vectors.

$$ \begin{align} \hat{b}_1 & = [1, 1, 1, 1] \frac{1}{2} \\ \hat{b}_2 & = [1, 0, -1, 0] \frac{1}{\sqrt{2}} \\ \hat{b}_3 & = [0, 1, 0, -1] \frac{1}{\sqrt{2}} \\ \hat{b}_4 & = [1, -1, 1, -1] \frac{1}{2} \\ \end{align} $$

And here is a vector.

$$ \begin{align} \vec{c} & = c_1 \hat{a}_1 + c_2 \hat{a}_2 + c_3 \hat{a}_3 + c_4 \hat{a}_4 \\ & = [c_1, c_2, c_3, c_4] \\ & = [5, 1, 7, -3] \\ \end{align} $$

1

Show by direct calculation using the dot product that each of these sets is orthonormal, which means that they are all length one and that any two different ones from the same set are perpendicular, i.e.

$$ \hat{x}_m \cdot \hat{x}_n =\begin{cases} 1 & \text{if} & m = n \\ 0 & \text{if} & m \ne n \\ \end{cases} $$

where

$$ x \, \text{is either} \, a \, \text{or} \, b \\ m, n \, \text{are each one of } \, 1,2,3,4 \\ $$

2

Find the coeficients $d_i$ of $\vec{c}$ in the $\hat{b}$ basis. In other words, find the four d's such that

$$ \vec{c} = d_1 \hat{b}_1 + d_2 \hat{b}_2 + d_3 \hat{b}_3 + d_4 \hat{b}_4 \\ $$

Hints:

• Use the dot product to find the component of some vector $\vec{h}$ in the direction of some basis vector $\hat{q}$.
• Use numpy arrays to define the dot product dot(x,y) as sum(array(x)*array(y)).