Exercise10

Given a set of linearly dependent vectors, we can eliminate some of the vectors to create a smaller, linearly independent set of vectors.

  1. Suppose that \(\wvec\) is a linear combination of the vectors \(\vvec_1\) and \(\vvec_2\text{.}\) Explain why \(\span{\vvec_1,\vvec_2, \wvec} = \span{\vvec_1,\vvec_2}\text{.}\)

  2. Consider the vectors

    \begin{equation*} \vvec_1 = \threevec{2}{-1}{0}, \vvec_2 = \threevec{1}{2}{1}, \vvec_3 = \threevec{-2}{6}{2}, \vvec_4 = \threevec{7}{-1}{1} \text{.} \end{equation*}

    Write one of the vectors as a linear combination of the others. Find a set of three vectors whose span is the same as \(\span{\vvec_1,\vvec_2,\vvec_3,\vvec_4}\text{.}\)

  3. Are the three vectors you are left with linearly independent? If not, express one of the vectors as a linear combination of the others and find a set of two vectors whose span is the same as \(\span{\vvec_1,\vvec_2,\vvec_3,\vvec_4}\text{.}\)

  4. Give a geometric description of \(\span{\vvec_1,\vvec_2,\vvec_3,\vvec_4}\) in \(\real^3\) as we did in Section 3.

in-context