Finding a Spanning Set for Polynomials of the Form (a+b+c)x^3+(a-2b)x^2+bx-c+a

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In summary, the spanning set for the vector space consisting of all polynomials of the given form is: {(x^3+x^2+1), (x^3+x^2+x), (x^3-1)} or equivalently {(1,0,1,1), (0,1,-2,1), (-1,0,0,1)}.
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Benzoate
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Homework Statement


Find a spanning set for the vector space consisting of all polynomials of the form:
(a+b+c)x^3+(a-2b)x^2+bx-c+a

Homework Equations


The Attempt at a Solution



a(1,0,1,1)+b(0,1,-2,1)+c(-1,0,0,1). So my spanning set is : {(1,0,1,1),(0,1,-2,1),(-1,0,0,1)}
 
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  • #2
If the space you want to span is made up of polynomials, your spanning set should be made of polynomials!
 
  • #3
quasar987 said:
If the space you want to span is made up of polynomials, your spanning set should be made of polynomials!

so my spanning set then is : span{(x^3+x^2+1),(x^3-2*x^2+x),(x^3-1)}
 
  • #4
Yes, that's fine.
 
  • #5
Yes, that's fine.
Another way of doing exactly the same thing:
You are given that the set of polynomials is all polynomials of the form (a+b+c)x^3+(a-2b)x^2+bx-c+a

Let a= 1, b= c= 0 and that is x^3+ x^2+ 1.
Let b= 1, a= c= 0 and that is x^3+ x^2+ x
Let c= 1, a= b= 0 and that is x^3- 1, exactly what you have.
 

1. How do I find a spanning set for polynomials of the form (a+b+c)x^3+(a-2b)x^2+bx-c+a?

In order to find a spanning set for polynomials of this form, you will first need to determine the possible values for the coefficients a, b, and c. Once you have determined the range of values for each coefficient, you can generate polynomials by choosing different combinations of these values and plugging them into the given equation.

2. What is a spanning set?

A spanning set is a set of vectors or polynomials that can be used to represent or generate all possible vectors or polynomials in a particular vector space. In other words, if a set of vectors or polynomials spans a vector space, then every vector or polynomial in that space can be written as a linear combination of those vectors or polynomials.

3. Why is finding a spanning set important?

Finding a spanning set is important because it allows us to represent a large or infinite set of vectors or polynomials in a more concise and manageable way. It also helps us to understand the structure and properties of a vector space or polynomial space by identifying the minimum number of vectors or polynomials needed to span that space.

4. What is the process for finding a spanning set?

The process for finding a spanning set involves determining the number of vectors or polynomials needed to span the given space, and then finding a set of vectors or polynomials that satisfy this requirement. This can be done by trial and error, by using a systematic method such as Gaussian elimination, or by using other techniques specific to the given problem.

5. Can there be more than one spanning set for a given vector space or polynomial space?

Yes, there can be more than one spanning set for a given vector space or polynomial space. This is because there are often multiple ways to represent a set of vectors or polynomials in a vector space, and therefore multiple ways to generate a spanning set. However, all spanning sets for a particular space will have the same number of elements.

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