Exercise 3

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

1. Find the linear combination with weights $c_1 = 2$, $c_2=-3$, and $c_3=1$.

2. Can you write the vector ${\mathbf 0} = \left[\begin{array}{r} 0 \\ 0 \end{array}\right]$ 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. Can you write the vector ${\mathbf 0} = \left[\begin{array}{r} 0 \\ 0 \end{array}\right]$ as a linear combination using just the first two vectors $\vvec_1$ $\vvec_2$? If so, describe all the ways in which you can do so.

4. Can you write $\vvec_3$ as a linear combination of $\vvec_1$ and $\vvec_2$? If so, in how many ways?