##### Exercise5

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 $$x_3\text{.}$$

in-context