Linear Algebra - Polynomial Subsets of Subspaces

[kaitamasaki]

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 $$a_{2} + a_{1} + a_{0} = 2$$

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 $$a_{2}t^{2}$$, 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?

 

[Hurkyl]

Don't guess: appeal to definitions and theorems! They give you a specific list of things to check, right? You say "it must be closed under addition and multiplication". What does that actually mean?


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

a) The set of all vectors of the form a₂t² + a₁t¹ + a₀ (where a₂, a₁, and a₀ are scalar-valued dummy variables) for which a₁ = 0 and a₀ = 0.