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

  • Thread starter dmitriylm
  • Start date
  • Tags
    Subspace
In summary, 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?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?Unless the set is
  • #1
dmitriylm
39
2

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?
 
Physics news on Phys.org
  • #2
dmitriylm said:

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.
 
  • #3
Do I need to do anything with the conditions a0a3 – a1a2 = 0?
 
  • #4
Absolutely! That equation describes the elements of your set.
 
  • #5
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?
 
Last edited:
  • #6
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?
 
  • #7
Thanks a lot for the help!
 
  • #8
Mark44 said:
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).
 
  • #9
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.
 
  • #10
dmitriylm said:

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.
 

1. What is a subspace?

A subspace is a subset of a vector space that contains vectors that can be added together and multiplied by scalars, while still remaining within the original vector space.

2. What is P3?

P3 is the vector space of polynomials with degree less than or equal to 3, where the coefficients are real numbers. It is denoted as P3 = {a0 + a1x + a2x^2 + a3x^3 | a0, a1, a2, a3 ∈ ℝ}.

3. How can I tell if a set is a subspace of P3?

To determine if a set is a subspace of P3, you must check if it satisfies three conditions: closure under vector addition, closure under scalar multiplication, and contains the zero vector. If all three conditions are met, then the set is a subspace of P3.

4. What is the condition for the set {a0 + a1x + a2x^2 + a3x^3 | a0a3 - a1a2 = 0} to be a subspace of P3?

The set {a0 + a1x + a2x^2 + a3x^3 | a0a3 - a1a2 = 0} will be a subspace of P3 if and only if the zero polynomial, which is the polynomial with all coefficients equal to 0, is included in the set. This ensures that the set contains the zero vector and satisfies the closure under scalar multiplication condition.

5. Can you provide an example of a polynomial in this set that is not a subspace of P3?

One example of a polynomial that is not a subspace of P3 is 2 + 3x + 4x^2 + 5x^3. This polynomial does not satisfy the condition a0a3 - a1a2 = 0, as 2*5 - 3*4 ≠ 0. Therefore, it is not a subspace of P3.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
17K
  • Calculus and Beyond Homework Help
Replies
9
Views
3K
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
422
  • Calculus and Beyond Homework Help
Replies
15
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
4K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Atomic and Condensed Matter
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
25
Views
4K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
Back
Top