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(i) The set of all vectors in R^2 of the form (a, b) where b = a
(ii) The set of all 3 × 3 matrices that have the vector [-2 -3 3]T as an eigenvector
(iii) The set of all polynomials in P2 of the form a0 + a1 x + a2 x2 where a0 = a2^2
I'm pretty sure that i) is correct and iii) is wrong but I might be wrong. Not sure about ii)
For the one(s) you think are true, try to prove them. For instance, in (i) take two arbitrary vectors of the form $(a,b)$ where $b=a$, e.g., $(c,c)$ and $(d,d)$, and add them. Do you again get something of the same form?
For the one(s) you think are false, try to come up with a counterexample. You said you think (iii) is false, why is that? Can you come up with explicit polynomials of the form described so that their sum is not of that form?
For the one(s) you're unsure about, start by trying to prove them. If it looks like the proof won't work, try to come up with a counterexample instead to disprove it.
Set (ii) is closed under addition.
Note $\mathbf{v} = (-2,-3,3)^T$. If $A_1.\mathbf{v} = \lambda_1 \mathbf{v}$ and $A_2.\mathbf{v} = \lambda_2 \mathbf{v}$ then $$(A_1+A_2).\mathbf{v} = A_1.\mathbf{v} + A_2.\mathbf{v} = \lambda_1 \mathbf{v} + \lambda_2 \mathbf{v} = (\lambda_1 + \lambda_2) \mathbf{v}$$ hence $A_1+A_2$ has $\mathbf{v}$ as an eigenvector.
