Finding the basis for a set of polynomials (linear algebra)

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SUMMARY

The discussion focuses on finding a basis for the set of polynomials in P3 that satisfy the conditions P'(1)=0 and P''(2)=0. The general form of a polynomial in P3 is given as ax^3 + bx^2 + cx + d. The derivatives P' and P'' are expressed as 3ax^2 + 2bx + c and 6ax + 2b, respectively. By substituting the specified values into the derived equations, the user successfully determines the basis for the polynomial set.

PREREQUISITES
  • Understanding of polynomial functions and their derivatives
  • Familiarity with the concept of a basis in linear algebra
  • Knowledge of the polynomial space P3
  • Ability to solve systems of linear equations
NEXT STEPS
  • Study the properties of polynomial spaces, specifically P3
  • Learn about the implications of derivative conditions on polynomial functions
  • Explore methods for solving systems of linear equations
  • Investigate the concept of linear independence in the context of polynomial bases
USEFUL FOR

Students studying linear algebra, particularly those focusing on polynomial functions and their properties, as well as educators seeking to enhance their understanding of polynomial bases in P3.

RossH
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Hi. Thanks for the help.

Homework Statement


Find a basis for the set of polynomials in P3 with P'(1)=0 and P''(2)=0.

Homework Equations


P' is the first derivative, P'' is the second derivative.


The Attempt at a Solution


The general form of a polynomial in P3 is ax^3+bx^2+cx+d
Therefore, P' will have the form 3ax^2+2bx+c
and P'' will have the form 6ax + 2b
Plugging in the known values, the two equations will be:
3a+2b+c=0
12a+2b+0c=0

I just don't know where to go from there, how to find the basis. I understand the concept of a basis and how to find one for a set of matrices or vectors, but not with this. Any help would be greatly appreciated. Thank you.
 
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Never mind. solved.
 

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