Exercise 10
Given a set of linearly dependent vectors, we can eliminate some of the vectors to create a smaller, linearly independent set of vectors.
Suppose that \(\wvec\) is a linear combination of the vectors \(\vvec_1\) and \(\vvec_2\text{.}\) Explain why \(\span{\vvec_1,\vvec_2, \wvec} = \span{\vvec_1,\vvec_2}\text{.}\)

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 \(\span{\vvec_1,\vvec_2,\vvec_3,\vvec_4}\text{.}\)
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 \(\span{\vvec_1,\vvec_2,\vvec_3,\vvec_4}\text{.}\)
Give a geometric description of \(\span{\vvec_1,\vvec_2,\vvec_3,\vvec_4}\) in \(\real^3\) as we did in SectionÂ 2.3.