find z such that $\vec{u}$ and $\vec{v}$ are linearly independent The problem is given as, "Determine the values of z such that the vectors $\vec{u} = \pmatrix{-1\\z}$ and $\vec{u} = \pmatrix{z\\-1 + z}$ are linearly independent.
Here is my work...
$\pmatrix{-1 & z& \\ z& (-1 + z)& }$
$\pmatrix{-1 & z& \\ 0& (z^2+z-1)& }$
therefore, $z^2+z-1 \neq 0$
$z \neq \frac{1}{2}(-1-\sqrt{5})$
$z \neq \frac{1}{2}(\sqrt{5} + 1)$
Does that seem right?

$$
z^2 + z - 1 \ne 0 \Rightarrow \\ 
z^2 + z +\frac{1}{4} \ne \frac{5}{4} \Rightarrow \\
\left(z+\frac{1}{2}\right)^2 \ne \frac{5}{4} \Rightarrow\\
z \ne -\frac{1}{2} + \frac{\sqrt{5}}{2}, z \ne -\frac{1}{2} - \frac{\sqrt{5}}{2} 
$$
