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

  • Thread starter dmitriylm
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  • #1
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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?
 

Answers and Replies

  • #2
34,976
6,729

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
39
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Do I need to do anything with the conditions a0a3 – a1a2 = 0?
 
  • #4
34,976
6,729
Absolutely! That equation describes the elements of your set.
 
  • #5
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2
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
34,976
6,729
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
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Thanks a lot for the help!
 
  • #8
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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
34,976
6,729
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
828
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?

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.
 

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