Exercise 5

This exercise involves a simple Fourier transform, which will play an important role in the next section.

Suppose that we have the vectors $$\vvec_1=\threevec{1}{1}{1}, \vvec_2=\threevec{\cos\left(\frac\pi6\right)} {\cos\left(\frac{3\pi}6\right)} {\cos\left(\frac{5\pi}6\right)} \vvec_3=\threevec{\cos\left(\frac{2\pi}6\right)} {\cos\left(\frac{6\pi}6\right)} {\cos\left(\frac{10\pi}6\right)}$$.

1. Explain why $\bcal=\{\vvec_1,\vvec_2,\vvec_3\}$ is a basis for $\real^3$. Notice that you may enter $\cos\left(\frac\pi6\right)$ into Sage as cos(pi/6).

2. If $\xvec=\threevec{15}{15}{15}$, find $\coords{\xvec}{\bcal}$.

3. Find the matrices $P_{\bcal}$ and $P_{\bcal}^{-1}$. If $\xvec=\threevec{x_1}{x_2}{x_3}$ and $\coords{\xvec}{\bcal} = \threevec{c_1}{c_2}{c_3}$, explain why $c_1$ is the average of $x_1$, $x_2$, and $x_3$.