Given the following vectors $\vec{u}$ and $\vec{v}$,

Question

I dont really know how to start such a question. I was thinking about \begin{matrix} -1 & -4 & 1 &0&0&0 \\ 0 & 8 & 0&1&0&0 \\ -2 & 4 & 0&0&1&0 \\ -3&1&0&0&0&1 \\ \end{matrix} but I am not sure. Can someone give me an idea of how to start such a question.

Answer 1

For $z$, think of (one of) the simplest linear combinations of $u$ and $v$.

For $w$, you are thinking in the right direction: you can choose it to be an element of the standard basis - just take one and see if it works, otherwise try another.