How to find basis vectors for a+ ax^2+bx^4?

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Discussion Overview

The discussion focuses on finding basis vectors for the polynomial expression a + ax^2 + bx^4, specifically within the context of the vector space of polynomials of degree at most 4 (p4). Participants explore the formulation of a basis for this polynomial and its implications.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant proposes that the basis for the polynomial a + ax^2 + bx^4 can be expressed as {1 + x^2, x^4}.
  • Another participant notes that since there are two parameters (a and b), it is reasonable to have a basis consisting of two vectors, indicating that the polynomial is in the span of the proposed basis for any choice of a and b.
  • A later reply acknowledges the previous contributions with a simple affirmation.

Areas of Agreement / Disagreement

There appears to be some agreement on the structure of the basis, but the correctness of the proposed basis remains unverified, and no consensus is explicitly stated regarding its validity.

DhineshKumar
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I want to find basis for a+ax^2+bx^4 belong to p4.
I am getting the following result is it right?
=>a(1+x^2) + b(x^4)
=> basis ={1+x^2, x^4}

Is that right ? Please help me any help is appreciated.
 
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Since you only have two terms (a,b) it makes sense to have a basis with two vectors. This function will be in the span of those bases for any choice of a and b.
In the standard polynomial basis [1, x, x^2, x^3, x^4,x^5 ... ], this would be [ a, 0, a, 0, b,0,...].
 
Thank you so much.
 
yes
 

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