Exercise 2

Consider the vectors $$\vvec_1 = \threevec{2}{-1}{0}, \vvec_2 = \threevec{1}{2}{1}, \vvec_3 = \threevec{2}{-2}{3}$$.

1. Are these vectors linearly independent or linearly dependent?

2. Describe the $\laspan{\vvec_1,\vvec_2,\vvec_3}$.

3. Suppose that $\bvec$ is a vector in $\real^3$. Explain why we can guarantee that $\bvec$ may be written as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$.

4. Suppose that $\bvec$ is a vector in $\real^3$. In how many ways can $\bvec$ be written as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$?