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.

1. Suppose that $\wvec$ is a linear combination of the vectors $\vvec_1$ and $\vvec_2$. Explain why $\laspan{\vvec_1,\vvec_2, \wvec} = \laspan{\vvec_1,\vvec_2}$.

2. Consider the vectors $$\vvec_1 = \threevec{2}{-1}{0}, \vvec_2 = \threevec{1}{2}{1}, \vvec_3 = \threevec{-2}{6}{2}, \vvec_4 = \threevec{7}{-1}{1}$$. Write one of the vectors as a linear combination of the others. Find a set of three vectors whose span is the same as $\laspan{\vvec_1,\vvec_2,\vvec_3,\vvec_4}$.

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 $\laspan{\vvec_1,\vvec_2,\vvec_3,\vvec_4}$.

4. Give a geometric description of $\laspan{\vvec_1,\vvec_2,\vvec_3,\vvec_4}$ in $\real^3$ as we did in The span of a set of vectors.