Exercise 5

Consider the vectors $$\vvec_1 = \left[\begin{array}{r} 2 \\ -1 \\ -2 \end{array}\right], \vvec_2 = \left[\begin{array}{r} 0 \\ 3 \\ 1 \end{array}\right], \vvec_3 = \left[\begin{array}{r} 4 \\ 4 \\ -2 \end{array}\right].$$

1. Can you express the vector $\bvec=\left[\begin{array}{r} 10 \\ 1 \\ -8 \end{array}\right]$ as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$? 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 $\vvec_1$, $\vvec_2$, and $\vvec_3$? 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 $\vvec_2$.

4. Explain why any linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$, $$a\vvec_1 + b\vvec_2 + c\vvec_3,$$ can be rewritten as a linear combination of just $\vvec_1$ and $\vvec_2$.