# Linear Algebra - Polynomial Subsets of Subspaces

## Homework Statement

Which one of the following subsets of $$P_{2}$$ (degree of 2 or below) are subspaces?
a) $$a_{2}t^{2} + a_{1}t + a_{0}$$, where $$a_{1} = 0$$ and $$a_{0} = 0$$
b) $$a_{2}t^{2} + a_{1}t + a_{0}$$, where $$a_{1} = 2a_{0}$$
c) $$a_{2}t^{2} + a_{1}t + a_{0}$$, where $$a_{2} + a_{1} + a_{0} = 2$$

## The Attempt at a Solution

First of all I don't even know if the question means only ONE of the three choices is a subspace, or whether I have to decide whether each of them are subspaces or not.

I know that in order for a subset to be a subspace, it must be closed under addition and multiplication.

a) would only have $$a_{2}t^{2}$$, but doesn't have the other degrees... is it not a subspace then? It is closed under addition ($$(a_{2}+a_{2})t^{2})$$ and and multiplication. Is the question implying the coefficient a2 doesn't change? So you can't have (a2)t^2 + (-a2)t^2 = 0 right?

b) I have all the terms for each degree, so I'm guessing it is a subspace?

c) I have no clue how to do this one. My guess is that I can factor out the 2 and it becomes 2($$t^{2} + t + 1$$), but if I add two of these together, I would get 4($$t^{2} + t + 1$$), so don't know if that is still in the subspace... does the scalar in front matter or not?

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Hurkyl
Staff Emeritus
Is that really the exact statement of the problem? It looks very sloppy to me. What they mean is surely something like: