Exercise7

Provide a justification for your response to the following questions.

  1. Suppose that we have vectors in \(\real^8\text{,}\) \(\vvec_1,\vvec_2,\ldots,\vvec_{10}\text{,}\) whose span is \(\real^8\text{.}\) Can every vector \(\bvec\) in \(\real^8\) be written as a linear combination of \(\vvec_1,\vvec_2,\ldots,\vvec_{10}\text{?}\)

  2. Suppose that we have vectors in \(\real^8\text{,}\) \(\vvec_1,\vvec_2,\ldots,\vvec_{10}\text{,}\) whose span is \(\real^8\text{.}\) Can every vector \(\bvec\) in \(\real^8\) be written uniquely as a linear combination of \(\vvec_1,\vvec_2,\ldots,\vvec_{10}\text{?}\)

  3. Do the vectors

    \begin{equation*} \evec_1=\threevec{1}{0}{0}, \evec_2=\threevec{0}{1}{0}, \evec_3=\threevec{0}{0}{1} \end{equation*}

    span \(\real^3\text{?}\)

  4. Suppose that \(\vvec_1,\vvec_2,\ldots,\vvec_n\) span \(\real^{438}\text{.}\) What can you guarantee about the value of \(n\text{?}\)

  5. Can 17 vectors in \(\real^{20}\) span \(\real^{20}\text{?}\)

in-context