###### Exercise3

Consider the following vectors in $$\real^4\text{:}$$

\begin{equation*} \vvec_1 = \fourvec{1}{1}{1}{1}, \vvec_2 = \fourvec{0}{1}{1}{1}, \vvec_3 = \fourvec{0}{0}{1}{1}, \vvec_4 = \fourvec{0}{0}{0}{1}\text{.} \end{equation*}
1. Explain why $$\bcal=\{\vvec_1,\vvec_2,\vvec_3,\vvec_4\}$$ forms a basis for $$\real^4\text{.}$$

2. Explain how to convert $$\coords{\xvec}{\bcal}\text{,}$$ the representation of a vector $$\xvec$$ in the coordinates defined by $$\bcal\text{,}$$ into $$\xvec\text{,}$$ its representation in the standard coordinate system.

3. Explain how to convert the vector $$\xvec$$ into, $$\coords{\xvec}{\bcal}\text{,}$$ its representation in the coordinate system defined by $$\bcal\text{.}$$

4. If $$\xvec=\fourvec{23}{12}{10}{19}\text{,}$$ find $$\coords{\xvec}{\bcal}\text{.}$$

5. If $$\coords{\xvec}{\bcal}=\fourvec{3}{1}{-3}{-4}\text{,}$$ find $$\xvec\text{.}$$

in-context