Is P4 a Subspace and What is Its Dimension?

In summary: It is just a convention.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. It is just a convention.
  • #1
bugatti79
794
1

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 [itex](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[/itex]

[itex]kp(x)=(kp)(x)=ka_0+ka_1x+ka_2x^2+ka_3x^3+ka_4x^4[/itex]

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
 
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  • #2
bugatti79 said:
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?
 
  • #3
Hurkyl said:
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)##...?
 
  • #4
bugatti79 said:
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.
 

FAQ: Is P4 a Subspace and What is Its Dimension?

1. What is the definition of a polynomial?

A polynomial is an algebraic expression consisting of variables and coefficients, with operations of addition, subtraction, and multiplication. It can include multiple terms, each with a variable raised to a non-negative integer power.

2. How is the degree of a polynomial determined?

The degree of a polynomial is determined by the highest power of the variable in the expression. For example, if a polynomial has the term 3x4, its degree is 4.

3. Can a polynomial have more than one variable?

Yes, a polynomial can have more than one variable. However, each term in a polynomial must have the same variables, and the variables must have non-negative integer exponents.

4. What operations can be performed on polynomials?

Polynomials can be added, subtracted, and multiplied with other polynomials. They can also be divided by another polynomial, but the resulting quotient may not always be a polynomial.

5. How are polynomials used in real-world applications?

Polynomials are used in a variety of fields, including physics, engineering, and economics. They can be used to model and solve problems involving quantities that change over time, such as population growth or the trajectory of a projectile.

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