Exercise5

Consider the vectors

\begin{equation*} \vvec_1 = \left[\begin{array}{r} 2 \\ -1 \\ -2 \end{array}\right], \vvec_2 = \left[\begin{array}{r} 0 \\ 3 \\ 1 \end{array}\right], \vvec_3 = \left[\begin{array}{r} 4 \\ 4 \\ -2 \end{array}\right]. \end{equation*}

  1. Can you express the vector \(\bvec=\left[\begin{array}{r} 10 \\ 1 \\ -8 \end{array}\right]\) as a linear combination of \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\text{?}\) If so, describe all the ways in which you can do so.

  2. Can you express the vector \(\bvec=\left[\begin{array}{r} 3 \\ 7 \\ 1 \end{array}\right]\) as a linear combination of \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\text{?}\) If so, describe all the ways in which you can do so.

  3. Show that \(\vvec_3\) can be written as a linear combination of \(\vvec_1\) and \(\vvec_2\text{.}\)

  4. Explain why any linear combination of \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\text{,}\)

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

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

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