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## Homework Statement

Which one of the following subsets of [tex]P_{2}[/tex] (degree of 2 or below) are subspaces?

a) [tex]a_{2}t^{2} + a_{1}t + a_{0}[/tex], where [tex] a_{1} = 0[/tex] and [tex]a_{0} = 0[/tex]

b) [tex]a_{2}t^{2} + a_{1}t + a_{0}[/tex], where [tex]a_{1} = 2a_{0}[/tex]

c) [tex]a_{2}t^{2} + a_{1}t + a_{0}[/tex], where [tex]a_{2} + a_{1} + a_{0} = 2[/tex]

## Homework Equations

## 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 [tex]a_{2}t^{2}[/tex], but doesn't have the other degrees... is it not a subspace then? It is closed under addition ([tex](a_{2}+a_{2})t^{2})[/tex] 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([tex]t^{2} + t + 1[/tex]), but if I add two of these together, I would get 4([tex]t^{2} + t + 1[/tex]), so don't know if that is still in the subspace... does the scalar in front matter or not?

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