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

In summary, to find a basis for the set of polynomials in P3 with P'(1)=0 and P''(2)=0, use the equations 3a+2b+c=0 and 12a+2b=0 formed by plugging in the known values for P' and P''. The basis will be the set of polynomials in the form ax^3+bx^2+cx+d that satisfy these equations.
  • #1
RossH
75
0
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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  • #2
Never mind. solved.
 

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

1. "What is the basis for a set of polynomials?"

The basis for a set of polynomials is the smallest set of polynomials that can be used to generate all other polynomials in the set through linear combinations. It is a fundamental concept in linear algebra and is used to simplify computations and understand the structure of polynomial spaces.

2. "How do you find the basis for a set of polynomials?"

To find the basis for a set of polynomials, you can use the method of Gaussian elimination or row reduction to put the polynomials into a matrix form. Then, you can use linear independence tests, such as the pivot columns or rank, to determine the number of basis polynomials. Finally, you can find the actual basis polynomials by solving the system of equations formed by the linearly independent polynomials.

3. "What is the significance of finding the basis for a set of polynomials?"

Finding the basis for a set of polynomials allows us to represent all polynomials in the set in a simpler and more systematic way. It also helps us to understand the structure of the polynomial space and make computations and proofs more efficient.

4. "Can there be more than one basis for a set of polynomials?"

Yes, there can be more than one basis for a set of polynomials. However, all bases for a given set of polynomials will have the same number of polynomials, known as the dimension of the polynomial space.

5. "How is finding the basis for a set of polynomials related to linear independence?"

Finding the basis for a set of polynomials is closely related to linear independence, as the basis polynomials are always linearly independent. In fact, one of the methods for finding the basis involves checking for linear independence. Additionally, the number of basis polynomials is equal to the rank of the polynomial set, which is a measure of linear independence.

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