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

1. ## 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

2. Originally Posted by osodud
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?

3. 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.