1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: How to find Insomorphism

  1. Aug 30, 2010 #1
    1. The problem statement, all variables and given/known data

    (i) Set of all row vectors: (a1,....,an), aj in K; addition, multiplication defined componentwise. This space is denoted as Kn.
    (ii) Set of all real valued functions f(x) defined on the real line, K = R.
    (iii) Set of all functions with values in K, defined on an arbitrary set S.
    (iv) Set of all polynomials of degree less than n with coefficients in K.

    2. Relevant equations
    1) Show that (i) and (iv) are isomorphic
    2) Show that if S has n elements, (i) is the same as (iii)
    3) Show that when K = R, (iv) is isomorphic with (iii) when S consists of n distinct points of R.

    3. The attempt at a solution
    I've solved 1), but I cannot solve others. I think that problem is that I don't understand definition of (iii).

    Could someone please help me?
  2. jcsd
  3. Aug 31, 2010 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    An example of (iii). If S is the rational numbers, then (iii) would be the set of functions from the rational numbers to the real numbers. Examples would be f(x)=x, f(x)=x2, f(x)=sin(x), where x is a rational number

    If S is the set containing just the numbers 1,4,7 and 9, then f(x) only takes four values. Because you only have f(1), f(4), f(7) and f(9). So if K is the real numbers again, a sample element of S would be the function f(x) with f(1)=2, f(4)=pi, f(7)=0 and f(9)=pi
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook