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 \(\coords{\xvec}{\bcal}\text{.}\)