The set P of polynomials

[bugatti79]

Homework Statement

Consider the set P4 of all real polynomials if degree <= 4.

1)Prove that P4 is a subspace of the vector space of all real polynomials
2)What is the dimension of the vector space P4. Prove answer by demonstrating a basis and verifying the proposed set is really a basis.

Homework Equations

The Attempt at a Solution

1)Let the vector $V = P={a_0+a_1x+a_2x^2+...+a_nx^n}$ where the coefficients are real numbers

let $p(x)=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4$ $q(x)=b_0+b_1x+b_2x^2+b_3x^3+b_4x^4$

Then $(p+q)(x) =p(x)+q(x)= (a_0+b_0)+(a_1+b_1)x+(a_2+b_2)x^2+(a_3+b_3)x^3+(a_4+b_4)x^4$

$kp(x)=(kp)(x)=ka_0+ka_1x+ka_2x^2+ka_3x^3+ka_4x^4$

thus p+q and kp are in V...?

2) Dimension is 5, ie we need 5 coefficients to have the basis linearly independent.
How do I demonstrate a basis and verify it?

Thanks

 

[Hurkyl]

2) Dimension is 5, ie we need 5 coefficients to have the basis linearly independent.
How do I demonstrate a basis and verify it?

It sounds like you already have a guess how to find the coordinates of a polynomial with respect to a particular basis; you just haven't yet found the basis. What would the coordinates of a basis vector be?

 

[bugatti79]

It sounds like you already have a guess how to find the coordinates of a polynomial with respect to a particular basis; you just haven't yet found the basis. What would the coordinates of a basis vector be?

Well the definition of a basis is that the basis set has to be linearly independent and it spans the vector V
So could we use $(1,x,x^2,x^3,x^4)$ since this is linearly independent.

Would the coordinates be the coefficients of $(a_0,a_1x,a_2x^2,a_3x^3,a_4x^4)$...?

 

[conquest]

Well the definition of a basis is that the basis set has to be linearly independent and it spans the vector V
So could we use $(1,x,x^2,x^3,x^4)$ since this is linearly independent.

Would the coordinates be the coefficients of $(a_0,a_1x,a_2x^2,a_3x^3,a_4x^4)$...?

As you say the basis is a set. It is customary to use { } to denote a set. Take care not to make the assumption that you absolutely need a (... ,... , ) notation to denote vectors in this case c+bx+cx²+ ... is a vector.