How to check if a polynomial equation belongs to a span

In summary, determining if p(x) is in the span of S involves solving a linear system of equations. If it is in the span, it can be expressed as a linear combination of the vectors in the set. Otherwise, it is not in the span.
  • #1
alexngo
4
0
hey
i would greatly appreciate the solutions for the question below

1) determine if p(x) = 9 - 17x + x^2 belong to the span of S {4-x+3x62, 2+5x+x^2}. If it does, express one vector as a linear combination of others. otherwise , justify your answer .
 
Physics news on Phys.org
  • #2
alexngo said:
hey
i would greatly appreciate the solutions for the question below

1) determine if p(x) = 9 - 17x + x^2 belong to the span of S {4-x+3x62, 2+5x+x^2}. If it does, express one vector as a linear combination of others. otherwise , justify your answer .

If p(x) is in the span of S then p(x)=a(4-x+3x62)+b(2+5x+x^2). Equate coefficients of the polynomial and solve the linear system of equations for the unknowns a and b.
 
  • #3
In general, a given vector is in the span of some set of vectors is a linear combination of the vectors in the set.

If you are working with a function space (as you are) rather than a vector space, replace the word "vector" in the previous sentence with "function."

The term "linear combination" of things in a set means a sum of scalar multiples of the things in the set.
 

1. What is a span in relation to a polynomial equation?

A span refers to the set of all possible linear combinations of a given set of vectors. In the context of a polynomial equation, it refers to the set of all possible polynomials that can be created using a given set of basis polynomials.

2. How do I determine if a polynomial equation belongs to a span?

To check if a polynomial equation belongs to a span, you can use the following steps:

  • Identify the basis polynomials for the span.
  • Write the given polynomial equation as a linear combination of the basis polynomials.
  • If the coefficients of the basis polynomials in the linear combination match the coefficients of the given polynomial equation, then it belongs to the span. Otherwise, it does not belong to the span.

3. Can a polynomial equation belong to more than one span?

Yes, a polynomial equation can belong to more than one span. This is because a span can have multiple sets of basis polynomials that can create the same polynomial equation through different linear combinations.

4. What is the importance of checking if a polynomial equation belongs to a span?

Checking if a polynomial equation belongs to a span helps in understanding the relationship between different sets of polynomials. It also helps in determining if a given set of polynomials can act as a basis for a larger space of polynomials.

5. Are there any alternative methods for checking if a polynomial equation belongs to a span?

Yes, there are alternative methods for checking if a polynomial equation belongs to a span. One method is to use row reduction techniques to solve a system of equations created by equating the coefficients of the polynomial equation and the basis polynomials. Another method is to use the concept of linear independence to determine if the given polynomial equation lies in the span of a set of polynomials.

Similar threads

  • Calculus and Beyond Homework Help
Replies
18
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
3K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
559
  • Calculus and Beyond Homework Help
Replies
24
Views
673
  • Calculus and Beyond Homework Help
Replies
2
Views
3K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
641
Back
Top