Is {a0 + a1x + a2x^2 + a3x^3 | a0a3 - a1a2 = 0} a subspace of P3?

[dmitriylm]

Homework Statement

Is {a0 + a1x + a2x^2 + a3x^3 | a0a3 - a1a2 = 0} a subspace of P3? Why or why not?

*The digits should be in subscript.

How would I go about answering this?

 

[Mark44]

Homework Statement

Is {a0 + a1x + a2x^2 + a3x^3 | a0a3 - a1a2 = 0} a subspace of P3? Why or why not?

*The digits should be in subscript.

How would I go about answering this?

Let's call your set as described above S. There are three things you need to verify to say that S is a subspace of P₃:

1. The zero polynomial is in S.
2. If p₁ and p₂ are any two polynomials in S, then p₁ + p₂ is in S.
3. If c is any real constant and p₁ is in S, the cp₁ is also in S.

 

[dmitriylm]

Do I need to do anything with the conditions a0a3 – a1a2 = 0?

 

[Mark44]

Absolutely! That equation describes the elements of your set.

 

[dmitriylm]

How do I derive the elements of my set from the equation?

If I select a0=2, a1=3, a2=4, and a3=6, then these numbers meet the requirements of the equation.

Similarly, a0=-2, a1=-3, a2=-4 and a3=-6 would as well. Where do I go from there?

 

[Mark44]

Unless the set is not a subspace and you can find some functions in the set that don't satisfy the subspace requirements, you won't be able to pick specific numbers for the coefficients.

Let p₁(x) = a₀ + a₁x + a₂x² + a₃x³ and p₂(x) = b₀ + b₁x + b₂x² + b₃x³, where both functions are in your set.

Is p₁(x) + p₂(x) also in the set?
Is k*p₁(x) in the set for any constant k?

 

[dmitriylm]

Thanks a lot for the help!

 

[Robert1986]

Let's call your set as described above S. There are three things you need to verify to say that S is a subspace of P₃:

1. The zero polynomial is in S.
2. If p₁ and p₂ are any two polynomials in S, then p₁ + p₂ is in S.
3. If c is any real constant and p₁ is in S, the cp₁ is also in S.

He doesn't need to show (1); this follows directly from (3).

 

[Mark44]

The list can be shortened even more. All you really need to show is that cp₁ + p₂ is in the set for any constant c and any functions p₁ and p₂ in the set.

 

[Robert1986]

Homework Statement

Is {a0 + a1x + a2x^2 + a3x^3 | a0a3 - a1a2 = 0} a subspace of P3? Why or why not?

*The digits should be in subscript.

How would I go about answering this?

I think that it might just be the polynomial thing that is causing you troubles; is that accurate? If it is, could you prove that a set of vectors in R^n is a subspace of R^n. That is, what if I asked this:

Is {(a_0),(a_1),(a_2),(a_3) | a0a3 - a1a2 = 0} a subspace of R^4, could you solve it? If so, you might want to prove (or disprove) that this is a subspace of R^4, then you can literally just take your proof and translate it to P^3. Do you understand what I am saying? I'm saying that R^4 and P^3 are essentialy the same (that is, they are isomorphic.) You can't prove it in R^4 and turn this in, but you can use it to get some insights.