Exercise2

Consider the vectors

\begin{equation*} \vvec_1 = \threevec{2}{-1}{0}, \vvec_2 = \threevec{1}{2}{1}, \vvec_3 = \threevec{2}{-2}{3} \text{.} \end{equation*}
  1. Are these vectors linearly independent or linearly dependent?

  2. Describe the \(\span{\vvec_1,\vvec_2,\vvec_3}\text{.}\)

  3. Suppose that \(\bvec\) is a vector in \(\real^3\text{.}\) Explain why we can guarantee that \(\bvec\) may be written as a linear combination of \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\text{.}\)

  4. Suppose that \(\bvec\) is a vector in \(\real^3\text{.}\) In how many ways can \(\bvec\) be written as a linear combination of \(\vvec_1\text{,}\) \(\vvec_2\text{,}\) and \(\vvec_3\text{?}\)

in-context