1. ## Subspace of P_4.

Pet P4 be the set of all polynomials of degree less than four.
Is the set S of polynomials of even degree less than four a subspace of P4?

I think that it is; however my text says it isn't.

Here's why I think that it is.

let v be an element of the Set described. Then v is an even degree polynomial of deg less than four. If we take a scaler œ and multiply v by it, we get another polynomial of the same deg before. So, œv is in S. Satisfying closure one.

Let v,x be in S, then consider v+x v+x is a polynomial with the highest even degree between them, and this will be less than 4, and it will be even. So, v+x is in S.

What did I do wrong, if anything?

2. Originally Posted by Chris11
Pet P4 be the set of all polynomials of degree less than four.
Is the set S of polynomials of even degree less than four a subspace of P4?

I think that it is; however my text says it isn't.

Here's why I think that it is.

let v be an element of the Set described. Then v is an even degree polynomial of deg less than four. If we take a scaler œ and multiply v by it, we get another polynomial of the same deg before. So, œv is in S. Satisfying closure one.

Let v,x be in S, then consider v+x v+x is a polynomial with the highest even degree between them, and this will be less than 4, and it will be even. So, v+x is in S.

What did I do wrong, if anything?
What is the degree of the zero polynomial ?

3. zero, but 0=2x0, so zero is an even number

4. I guess that x^2-2x is in P4 and -x^2 is as well. But xthe sum of those 2 polynomials is -2x, and that's not even. So, I guess I sort of rushed it

5. Originally Posted by Chris11
zero, but 0=2x0, so zero is an even number
Yes, but that describes all polynomials of the form $\displaystyle a_0 + 0x + 0x^2 + 0x^3$. The zero vector here is the zero polynomial which is defined to have degree of $\displaystyle -\infty$.

Either way, I'm sure you can see that our set doesn't satisfy the conditions for it to be a subspace of $\displaystyle \mathcal{P}_4$

6. Originally Posted by o_O
Yes, but that describes all polynomials of the form $\displaystyle a_0 + 0x + 0x^2 + 0x^3$. The zero vector here is the zero polynomial which is defined to have degree of $\displaystyle -\infty$.
Can you give a reference for that? Every book I have ever seen says that the zero polynomial has degree 0.

Either way, I'm sure you can see that our set doesn't satisfy the conditions for it to be a subspace of $\displaystyle \mathcal{P}_4$
Well, actually, the reason for posting the question in the first place was that he couldn't!

7. Well, actually, I can. I didn't know that the degree of the zero polynomial was defined as -infinity. lol

8. Originally Posted by HallsofIvy
Can you give a reference for that? Every book I have ever seen says that the zero polynomial has degree 0.
I was taught that it was. The textbook I used was Axler's Linear Algebra Done Right. And wiki seems to agree too.

But for a perhaps better reference, here's an entry about the zero polynomial on Wolfram: Zero Polynomial

Looks like it's either undefined, $\displaystyle -\infty$, or $\displaystyle -1$. I'm pretty sure it can't be 0 though .. not that I'm an expert or anything.

Well, actually, the reason for posting the question in the first place was that he couldn't!
I meant that through the post about the degree of the zero polynomial and the OP's post which contained a counterexample, it should all be clear now.

9. you can also find that the zero polynomial has no degree.

10. Originally Posted by o_O
I was taught that it was. The textbook I used was Axler's Linear Algebra Done Right. And wiki seems to agree too.

But for a perhaps better reference, here's an entry about the zero polynomial on Wolfram: Zero Polynomial

Looks like it's either undefined, $\displaystyle -\infty$, or $\displaystyle -1$. I'm pretty sure it can't be 0 though .. not that I'm an expert or anything.

I meant that through the post about the degree of the zero polynomial and the OP's post which contained a counterexample, it should all be clear now.
Thank you!