Register to reply

Two vector spaces being isomorphic

by terryfields
Tags: isomorphic
Share this thread:
terryfields
#1
Jan5-09, 11:03 AM
P: 44
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????
Phys.Org News Partner Science news on Phys.org
Bees able to spot which flowers offer best rewards before landing
Classic Lewis Carroll character inspires new ecological model
When cooperation counts: Researchers find sperm benefit from grouping together in mice
Big-T
#2
Jan5-09, 05:21 PM
P: 64
How many basis vectors do you need to span P_k?
HallsofIvy
#3
Jan5-09, 07:53 PM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,348
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.

terryfields
#4
Jan6-09, 08:21 AM
P: 44
Two vector spaces being isomorphic

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?
Dick
#5
Jan6-09, 08:31 AM
Sci Advisor
HW Helper
Thanks
P: 25,244
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.
terryfields
#6
Jan7-09, 11:58 AM
P: 44
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)
Dick
#7
Jan7-09, 01:33 PM
Sci Advisor
HW Helper
Thanks
P: 25,244
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.
terryfields
#8
Jan7-09, 05:09 PM
P: 44
thanks, thats much simpler than it first looked.


Register to reply

Related Discussions
Prove not isomorphic? Linear & Abstract Algebra 2
Isomorphic groups? Linear & Abstract Algebra 3
Isomorphic Help! Calculus & Beyond Homework 4
Is this number isomorphic? Calculus & Beyond Homework 4
Isomorphic: PLease Help Calculus & Beyond Homework 7