Let 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 ?
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 . If V also has dimension n, then it has a basis . The function, , defined by and extended "by linearity" is an isomorphism.