Exercise5

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

Suppose that we have the vectors

\begin{equation*} \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)} \text{.} \end{equation*}
  1. Explain why \(\bcal=\{\vvec_1,\vvec_2,\vvec_3\}\) is a basis for \(\real^3\text{.}\)

  2. If \(\xvec=\threevec{15}{15}{15}\text{,}\) find \(\coords{\xvec}{\bcal}\text{.}\)

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

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