Exercise10

Suppose that

\begin{equation*} \vvec_1 = \fourvec{3}{1}{3}{-1}, \vvec_2 = \fourvec{0}{-1}{-2}{2}, \vvec_3 = \fourvec{-3}{-3}{-7}{5} \text{.} \end{equation*}
  1. Is \(\vvec_3\) a linear combination of \(\vvec_1\) and \(\vvec_2\text{?}\) If so, find weights such that \(\vvec_3 = a\vvec_1+b\vvec_2\text{.}\)

  2. Show that a linear combination

    \begin{equation*} a\vvec_1 + b\vvec_2 + c\vvec_3 \end{equation*}

    can be rewritten as a linear combination of \(\vvec_1\) and \(\vvec_2\text{.}\)

  3. Explain why \(\span{\vvec_1,\vvec_2,\vvec_3} = \span{\vvec_1,\vvec_2}\text{.}\)

in-context