Exercise3

Consider the vectors

\begin{equation*} \vvec_1 = \left[\begin{array}{r} 2 \\ 1 \end{array} \right], \vvec_2 = \left[\begin{array}{r} -1 \\ 1 \end{array} \right], \vvec_3 = \left[\begin{array}{r} -2 \\ 0 \end{array} \right] \end{equation*}
  1. Find the linear combination with weights \(c_1 = 2\text{,}\) \(c_2=-3\text{,}\) and \(c_3=1\text{.}\)

  2. Can you write the vector \({\mathbf 0} = \left[\begin{array}{r} 0 \\ 0 \end{array}\right]\) 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. Can you write the vector \({\mathbf 0} = \left[\begin{array}{r} 0 \\ 0 \end{array}\right]\) as a linear combination using just the first two vectors \(\vvec_1\) \(\vvec_2\text{?}\) If so, describe all the ways in which you can do so.

  4. Can you write \(\vvec_3\) as a linear combination of \(\vvec_1\) and \(\vvec_2\text{?}\) If so, in how many ways?

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