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

[Benzoate]

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)}

 

[quasar987]

If the space you want to span is made up of polynomials, your spanning set should be made of polynomials!

 

[Benzoate]

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)}

 

[HallsofIvy]

Yes, that's fine.

 

[HallsofIvy]

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.