[Math] Determining if a polynomial of a form is a subspace of all 3rd degree polynomails

linear algebravector-spaces

I am suppose to determine if all polynomials of the form $a_0 + a_1x +a_2x^2+a_3x^3$, where $a_0 = 0$ is a subspace of all 3rd degree polynomials.

I'm pretty sure I know the answer, but I'm just looking for verification of my understanding.

  1. I first check to see if the zero vector belongs to the polynomial form given, indeed it does.

  2. Whenever p(x) and q(x) belongs to the given polynomial form, then the addition of them must as well.

    • This is where it fails I believe, for instance if you had p(x) where the 3rd degree variable is $5x^3$ and q(x) is $-5x^3$, then in the total summation (whatever the other variables are), the total result would be of some form $0+ a_1x +a_2x^2+0$. Which can't be a subspace of $a_0 + a_1x +a_2x^2+a_3x^3$.

Is this the correct approach or am I misunderstanding anything?

Edit: Seeing as it is actually a subspace, anything with a degree equal to or less than 3 would be a subspace?

Best Answer

The condition (from the way you've worded it) is just $a_0 = 0$ so even if $a_3 = 0$, you still belong in the subspace. So, I believe this is a subspace. A basis for this subspace is given by $\{x, x^2, x^3\}$. Now if the condition also included that $a_3 \not= 0$ then you are right, this is not a subspace.

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