Two vector spaces being isomorphic

In summary, the conversation discusses the question of determining the value of n for which the vector space Pk, consisting of real polynomials of degree at most k, is isomorphic to Rn. The conclusion is that the two vector spaces are isomorphic if they have the same dimension, with Pk having a dimension of k+1 and Rn having a dimension of n. This leads to the answer that Pk is isomorphic to Rn when n = k+1.
  • #1
terryfields
44
0
sadly not been able to put much effort into this one! was a lecture i missed towards the end of term and didnt get the notes on it, however here is the question.
for K>or equal to 1 let Pk denote the the vector space of all real polynomials of degree at most k. For which value of n is Pk isomorphic to Rn. Give a brief reason for your answer.

Now from what i have found on two vector spaces being isomorphic they need to have equal dimensions (dimu=dimv) so knowing that we have dimRn)=n is as far as i have got. Not really understanding this one, surely they would be isomorphic at any value of n as long as it's between 1 and k?
 
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  • #2


How many basis vectors do you need to span P_k?
 
  • #3


For example, P1 is the space of all first degree polynomials which can be written in the form ax+ b which, in turn, can be mapped to (a,b) in R2.
 
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  • #4


so for a second degree polynomial you would have ax2 +bx +c so is the answer just n-1 because the dimension of a polynomial is always one higher than the degree?
 
  • #5


A single polynomial doesn't have a 'dimension'. The point is that the set of ALL degree two or less polynomials can be represented as linear combinations of the three linearly independent functions 1, x and x^2. They form a basis. The 'dimension' of a space is the number of elements in a basis.
 
  • #6


Ok, so the basis formed by the polynomial is always going to be one higher than the degree of the polynomial, therefore value of n that will make Pk isomorphic to n has to be k+1? (crosses fingers and preys)
 
  • #7


Preys? I think you want to pray. Sure, P_k has dimension k+1. R^n has dimension n. Two finite dimensional vector spaces are isomorphic if they have the same dimension.
 
  • #8


thanks, that's much simpler than it first looked.
 

1. What does it mean for two vector spaces to be isomorphic?

Two vector spaces are isomorphic if there exists a linear transformation between them that is both one-to-one and onto. This means that the two vector spaces have the same dimension and structure, and can essentially be considered the same space.

2. How can you prove that two vector spaces are isomorphic?

To prove that two vector spaces are isomorphic, you must show that there exists a bijective linear transformation between them. This can be done by demonstrating that the transformation is both one-to-one and onto, and that it preserves vector addition and scalar multiplication.

3. Are all vector spaces isomorphic?

No, not all vector spaces are isomorphic. In order for two vector spaces to be isomorphic, they must have the same dimension and structure. Therefore, vector spaces with different dimensions or structures cannot be isomorphic.

4. How are isomorphic vector spaces useful in mathematics?

Isomorphic vector spaces are useful in mathematics because they allow us to simplify and generalize concepts. By showing that two vector spaces are isomorphic, we can apply theorems and properties from one space to the other, making problem solving and proof writing easier.

5. Can two vector spaces be isomorphic even if they have different bases?

Yes, two vector spaces can be isomorphic even if they have different bases. This is because the basis of a vector space is not what determines its structure, but rather its dimension and the relationships between its elements. As long as these properties are preserved by the linear transformation, the vector spaces can be considered isomorphic.

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