Proving a set of non-linear equations are a subspace in ${R}^3$

Question

For each value of the parameter 'a' determine whether the set of solutions of the following system of equations,

$$x + (1-{a}^2){y}^3 - 2z = 0$$ $$x + y - (2{a}^2 + 2a)|z| = {a}^3 - a$$

is a subspace in $R^3$ or not.

Since this system is not linear I can't make an augmented matrix to satisfy the subspace conditions. How do I start?

Answer 1

Hint: If the following set of solutions is a subspace, $(0,0,0)$ must be a solution and thus $a^3=a$. Now you only have $3$ cases to consider.