1. ## Subspaces and Bases

This problem seems like a trick question, I am not sure how to approach it.

Is the following a subspace? If so, find the bases and dimension.
P ∈ R[x] ; P^2 = 0

I know that it is a subspace if:
The zero vector is included in the set
Closed under scalar multiplication

From first glance it seems like a subspace but how do I prove this and would it have a 0 basis?

2. Originally Posted by natcapat
QAlop{:"}]=-091`qThis problem seems like a trick question and I am not quite sure how to approach it. The simplest questions really throw me off.

Is the following a subspace?
}"{p/;olikmujnbgveaq
Sorry - what is the question...?

(you might want to use LaTeX, but use [tex ] and [/tex ] instead of [math ] and [/math ]).

3. Originally Posted by natcapat
This problem seems like a trick question, I am not sure how to approach it.

Is the following a subspace? If so, find the bases and dimension.
P ∈ R[x] ; P^2 = 0

I know that it is a subspace if:
The zero vector is included in the set
Closed under scalar multiplication

From first glance it seems like a subspace but how do I prove this and would it have a 0 basis?
If by

$\displaystyle R[x] \text{ You mean } \mathbb{R}[x]$

Then since the real numbers are a field the polynomial ring must be an integral domain.
If R is just a Ring the we will need some information about the units of R.

4. Originally Posted by TheEmptySet
If by

$\displaystyle R[x] \text{ You mean } \mathbb{R}[x]$

Then since the real numbers are a field the polynomial ring must be an integral domain.
If R is just a Ring the we will need some information about the units of R.
..ignore this post...I was wrong...

5. That is my problem I wasn't given any more information than just that. I understand that it is all real number in the field of polynomials but I do not know what do after. The problem given was exactly this:

$P \epsilon \mathbb{R}[x]: P^{^{2}} = 0$

6. Originally Posted by natcapat
That is my problem I wasn't given any more information than just that. I understand that it is all real number in the field of polynomials but I do not know what do after. The problem given was exactly this:

$P \epsilon \mathbb{R}[x]: P^{^{2}} = 0$
Well, what polynomials square to give 0? Hint: there aren't very many of them...! (TheEmptySet tells you why, above.)

7. Note that when we say a polynomial (or any function) is 0, we mean it is 0 for all x.

8. hint #2: such a polynomial must be constant (why?).