Activity3.2.2

  1. In the preview activity, we considered a set of vectors in \(\real^2\text{:}\)

    \begin{equation*} \vvec_1 = \twovec{2}{1}, \vvec_2 = \twovec{1}{2} \text{.} \end{equation*}

    Explain why these vectors form a basis for \(\real^2\text{.}\)

  2. Consider the set of vectors in \(\real^3\)

    \begin{equation*} \vvec_1 = \threevec{1}{1}{1}, \vvec_2 = \threevec{0}{1}{-1}, \vvec_3 = \threevec{1}{0}{-1} \text{.} \end{equation*}

    and determine whether they form a basis for \(\real^3\text{.}\)

  3. Do the vectors

    \begin{equation*} \vvec_1 = \threevec{-2}{1}{3}, \vvec_2 = \threevec{3}{0}{-1}, \vvec_3 = \threevec{1}{1}{0}, \vvec_4 = \threevec{0}{3}{-2} \end{equation*}

    form a basis for \(\real^3\text{?}\)

  4. Explain why the vectors \(\evec_1,\evec_2,\evec_3\) form a basis for \(\real^3\text{.}\)

  5. If a set of vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\) forms a basis for \(\real^m\text{,}\) what can you guarantee about the pivot positions of the matrix

    \begin{equation*} \left[\begin{array}{rrrr} \vvec_1 \amp \vvec_2 \amp \ldots \amp \vvec_n \end{array}\right] \text{?} \end{equation*}
  6. If the set of vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\) is a basis for \(\real^{10}\text{,}\) how many vectors must be in the set?

in-context