Prove/Disprove: Existence of a Basis of P_3(F) w/ Degree 2 Polynomial

In summary, the conversation discusses proving or disproving the existence of a basis of P_3(F) in which one of the polynomials has degree 2. The solution provided shows that such a basis does not exist and that the given set of polynomials forms a basis for P_3(F). A possible additional proof using a matrix is suggested. A typo in the original statement is also corrected.
  • #1
jaejoon89
195
0

Homework Statement



Prove or disprove: there exists a basis (p_0, p_1, p_2, p_3) of P_3 (F) such that one of the polynomials p_0, p_1, p_2, p_3 has degree 2.

Homework Equations



none really

The Attempt at a Solution



Is the following proof correct?

----

Let p_0, p_1, p_2, p_3 be elements of P_3(F) s.t.

p_o (x) = 1,
p_1 (x) = x,
p_2 (x) = x^2 + x^3,
p_3(x) = x^3.

None of the polynomials are degree 2 although (p_0,p_1,p_2,p_3) is clearly spanning P_3 (F) with dimP_3(F) = 4 and forms a basis. Hence proved.
 
Physics news on Phys.org
  • #2
Looks good to me assuming the "one" in the statement of the problem should be "none".
 
  • #3
I'm not sure that stating that it clearly spans it will suffice even if it is obvious. If you think that this suffices for you class, you're fine.

On the other hand, you could cook up a matrix that maps a degree three polynomial represented in the standard basis to it's representation in this basis pretty easily.
 
  • #4
Typo: the original statement is supposed to be "none of the polynomials has degree 2." Thanks for pointing that out, LCKurtz.
 

1. Can you explain what "P_3(F)" means in this context?

P_3(F) refers to the set of all polynomials with degree 3 or less and coefficients from the field F. Essentially, it is the set of all polynomials with 3 or less terms.

2. What does it mean for a basis to exist in P_3(F) with degree 2 polynomials?

A basis is a set of vectors that can be used to represent all other vectors in a vector space. In this case, a basis of P_3(F) with degree 2 polynomials means that there exists a set of degree 2 polynomials that can be used to represent all other polynomials with degree 3 or less in the field F.

3. How can the existence of a basis in P_3(F) with degree 2 polynomials be proven or disproven?

The existence of a basis in P_3(F) with degree 2 polynomials can be proven by showing that a set of degree 2 polynomials spans the entire space of polynomials with degree 3 or less in the field F. It can be disproven by showing that no such set exists.

4. Is there a specific method or algorithm for finding a basis in P_3(F) with degree 2 polynomials?

Yes, there are several methods for finding a basis in P_3(F) with degree 2 polynomials. One method is to use the Gram-Schmidt process, which involves orthogonalizing a set of polynomials to create a basis. Another method is to use linear algebra techniques, such as finding the null space of a matrix, to determine a basis.

5. What are the implications of proving or disproving the existence of a basis in P_3(F) with degree 2 polynomials?

If it is proven that a basis exists, it means that all polynomials with degree 3 or less in the field F can be uniquely represented by a combination of degree 2 polynomials. This has important implications in fields such as linear algebra, where bases are used to solve systems of equations. If it is disproven, it means that there is no set of degree 2 polynomials that can represent all polynomials with degree 3 or less in the field F, which could have implications in fields such as polynomial interpolation and approximation.

Similar threads

MHB Quadratic Polynomial Problem: Proving Equality with Real Numbers
    • Math POTW for Secondary and High School Students
Replies
1
Views
874
MHB Polynomial in n variables: Prove the identity
    • General Math
Replies
1
Views
1K
Is S a Subspace of P_3 and Does q(x) Belong in S?
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Implicitization via Moving Curves
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
I Existence of basis for P_2 with no polynomial of degree 1
    • Linear and Abstract Algebra
Replies
13
Views
1K
Direct Proof that every zero of p(T) is an eigenvalue of T
    • Calculus and Beyond Homework Help
Replies
24
Views
793
Introduction to Linear Algebra: Solving Real-World Problems
    • Calculus and Beyond Homework Help
Replies
10
Views
1K
Tricky Problem: Prove range T = null ##\phi## when null T' has dim 1
    • Calculus and Beyond Homework Help
Replies
8
Views
619
Prove that range ##T'## = ##(\text{null}\ T)^0##
    • Calculus and Beyond Homework Help
Replies
1
Views
456
Finding a complementary subspace ##U## | Linear Algebra
    • Calculus and Beyond Homework Help
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top