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 2.3.

in-context