##### Exercise5

Consider the vectors

\begin{equation*} \vvec_1 = \left[\begin{array}{r} 2 \\ -1 \\ -2 \end{array}\right], \vvec_2 = \left[\begin{array}{r} 0 \\ 3 \\ 1 \end{array}\right], \vvec_3 = \left[\begin{array}{r} 4 \\ 4 \\ -2 \end{array}\right]. \end{equation*}

1. Can you express the vector $$\bvec=\left[\begin{array}{r} 10 \\ 1 \\ -8 \end{array}\right]$$ as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3\text{?}$$ If so, describe all the ways in which you can do so.

2. Can you express the vector $$\bvec=\left[\begin{array}{r} 3 \\ 7 \\ 1 \end{array}\right]$$ as a linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3\text{?}$$ If so, describe all the ways in which you can do so.

3. Show that $$\vvec_3$$ can be written as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$

4. Explain why any linear combination of $$\vvec_1\text{,}$$ $$\vvec_2\text{,}$$ and $$\vvec_3\text{,}$$

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

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

in-context