Exercise10

Suppose that

\begin{equation*} \vvec_1 = \fourvec{3}{1}{3}{-1}, \vvec_2 = \fourvec{0}{-1}{-2}{2}, \vvec_3 = \fourvec{-3}{-3}{-7}{5}\text{.} \end{equation*}
1. Is $$\vvec_3$$ a linear combination of $$\vvec_1$$ and $$\vvec_2\text{?}$$ If so, find weights such that $$\vvec_3 = a\vvec_1+b\vvec_2\text{.}$$

2. Show that a linear combination

\begin{equation*} a\vvec_1 + b\vvec_2 + c\vvec_3 \end{equation*}

can be rewritten as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$

3. Explain why $$\span{\vvec_1,\vvec_2,\vvec_3} = \span{\vvec_1,\vvec_2}\text{.}$$

in-context