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Math Help - Vector space isomorphic to R^n+1

  1. #1
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    Question Vector space isomorphic to R^n+1

    Hello

    Let P_n (R)= \left\{{a_0 + a_1x + a_2x^2 + ... + a_nx^n:  a_i \in{R}}\right\} be our set.

    The sum and the scalar product are defined normally.

    How can i prove that this set is with those operations is isomorphic to  R^n^+^1 ?

    Thanks
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  2. #2
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    Quote Originally Posted by osodud View Post
    Hello

    Let P_n (R)= \left\{{a_0 + a_1x + a_2x^2 + ... + a_nx^n:  a_i \in{R}}\right\} be our set.

    The sum and the scalar product are defined normally.

    How can i prove that this set is with those operations is isomorphic to  R^n^+^1 ?

    Thanks
    Does the map

    T:\mathbb{P}_n \to \mathnn{R^{n+1}} defined by

    T(ax^j)=\underbrace{(0,0,...,a,0,...,0))}_{\text{ a is in the j+1 component}}
    Does this define an isomorphism?
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  3. #3
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    A very general and not very difficult theorem- if two finite dimensional vector spaces have the same dimension, then they are isomorphic.

    That is, if U has dimension n, then it has a basis \{u_1, u_2, \cdot\cdot\cdot, u_n\}. If V also has dimension n, then it has a basis \{v_1, v_2, \cdot\cdot\cdot, \v_n\}. The function, f:U\to V, defined by f(u_i)= v_i and extended "by linearity" is an isomorphism.
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