Math Help - Finding a basis for...

1. Finding a basis for...

This was a bonus question on my test yesterday (I hope I can remember it correctly).

Q: Let W be a subsapce of $P_{3}$ such that P(0)=0 and let U be a subspace of $P_{3}$ such that P(1)=0. Find a basis for both W and U.

A: For W I figured the basis ought to be $\{x,x^{2},x^{3}\}$ since I don't wan't $a_{0}$ to have any value. But, I am stuck on U. I think I am seeing it all wrong. My x's are my vectors correct? As in, $\{1,x,x^{2},x^{3}\}$ is my basis for all third degree polynomails which have the form $a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}$. I tried it a couple different ways on the exam, but should said they were all wrong, but that I was close. Even so, I think I am stuck in a rut and I am having a hard time starting from scratch. Im having trouble finding a systematic approach.

Thanks

2. Originally Posted by Danneedshelp
This was a bonus question on my test yesterday (I hope I can remember it correctly).

Q: Let W be a subsapce of $P_{3}$ such that P(0)=0 and let U be a subspace of $P_{3}$ such that P(1)=0. Find a basis for both W and U.

A: For W I figured the basis ought to be $\{x,x^{2},x^{3}\}$ since I don't wan't $a_{0}$ to have any value. But, I am stuck on U. I think I am seeing it all wrong. My x's are my vectors correct? As in, $\{1,x,x^{2},x^{3}\}$ is my basis for all third degree polynomails which have the form $a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}$. I tried it a couple different ways on the exam, but should said they were all wrong, but that I was close. Even so, I think I am stuck in a rut and I am having a hard time starting from scratch. Im having trouble finding a systematic approach.

Thanks
You are correct for V. For U, think about $\{x-1, x^2- 1, x^3- 1\}$, exactly like you have for V but "shifted".

For a more "systematic" method try this. A general vector in P(3) is $a+ bx+ cx^2+ dx^3$ and your subset is defined by $p(1)= a+ b+ c+ d= 0$. We can solve for any one of the coefficients as a function of the other 3 so this is a subspace of dimension 3. For example, a= -b- c- d. Now here is a very nice method for finding bases when you have a relation like that. Take b= 1, c= d= 0 and you get a= -1. That is, -1+ x or x-1 is in the set. Take c= 1, b= d= 0. Again a= -1 so we get $-1+x^2$. Finally, take d=1, b= c= 0. Yet again a= -1 so $-1+ x^3$. Taking each constant equal to 1 guarentees that those are independent so that is a basis, the same one I mentioned before.

Of course, we could have solved for any of the four coefficients. If, say, we had solved for d= -a- b- c, then: with a= 1, b=c= 0, we get $1- x^3$. With b=1, a= c= 0, we get $x- x^3$. With c= 1, a= b= 0, we get $x^2- x^3$ getting $\{1- x^3, x- x^3, x^2-x^3\}$ as a different basis for that subspace.

3. Originally Posted by Danneedshelp
This was a bonus question on my test yesterday (I hope I can remember it correctly).

Q: Let W be a subspace of $P_{3}$ such that P(0)=0 and let U be a subspace of $P_{3}$ such that P(1)=0. Find a basis for both W and U.
I was really confused by your question at first as I have never seen this notation before for polynomials and thought it was something to do with a projective space. Does anyone know if this is a standard notation?

Also it seems to me that the question as posed does not have an answer. Surely U and W should be the maximal subspaces with this property otherwise you can't tell whether they have full rank.

4. Originally Posted by alunw
I was really confused by your question at first as I have never seen this notation before for polynomials and thought it was something to do with a projective space. Does anyone know if this is a standard notation?

Also it seems to me that the question as posed does not have an answer. Surely U and W should be the maximal subspaces with this property otherwise you can't tell whether they have full rank.
Yes, that is standard notation and "P(n)", the vector space (or algebra) of polynomials of degree less than or equal to n, is a common example in Linear Algebra. And "maximal" would be understood here.

5. Thanks, but I must say I don't like either the notation or the convention that maximal would be understood.
There's a much better and more widely used notation for the space of all real polynomials: $\mathbb{R}[x]$ and some notation like $\mathbb{R}[x]_{3}$ would convey a lot more information. Personally I'd prefer a name like Poly(R,{x},3) since that would be something one could adapt for a computer program and have something much more likely to be meaningful to someone coming across things for the first time.
What's the benefit of writing "a subspace" instead of "the maximal subspace"? To my way of thinking it's going to lead to one assuming a subspace is maximal when its not meant to be. In group theory homomorphisms are almost always surjective, but plenty of writers still insert the word surjective rather than saying this is to be understood.

6. Thank you very much HallsofIvy!

Ugh, I'm kickin myself already! Makes perfect sense. I was just getting confused with the notation and thought each a,b,c, and d was distributed through every vector. I don't know why I was thinking that when I new what the general form was.

7. All cases discussed before :

I have tried to solve for variables other than 'a' and have obtained the results. I have attached that as an image. Hope it helps.