What values of $\lambda$ make $\{u, v, w\}$ linearly dependent?
You made some mistake while solving the system, because its solutions are $(\alpha,\beta,\lambda)=(1,0,0)$, $(\alpha,\beta,\lambda)=\left(-1,\sqrt2,\sqrt2\right)$, and $(\alpha,\beta,\lambda)=\left(-1,-\sqrt2,-\sqrt2\right)$. Besides, this will give you only the cases in which $\vec w$ is a linear combination of $\vec u$ and $\vec v$. But it could also happen that $\vec u$ is a linear combination of $\vec v$ and $\vec w$ or that $\vec v$ is a linear combination of $\vec u$ and $\vec w$.