Linear algebra, proof that it's an isomorphism.

Question

I'm dealing with isomorphisms, and I'm not quite sure of how to formulate this one. Mainly the elements from each function, so I can start to proof that is a linear and a bijective function. Here are the details:

If $V$ is the vector space of all polynomials of degree less than or equal to $n$, with coefficients in the field $K$. Proof that.

$V\simeq K^{n+1}$

I'll be thankful if any help can be given.

Answer 1

Hint: Send $a_0+a_1x+a_2x^2+\cdots+a_nx^n \in V$ to $(a_0,a_1,a_2,\ldots,a_n) \in K^{n+1}$.

Answer 2

Hint: the coefficients of a polinomial of degree less or equal than n are $n+1$, and they completely determine the polinomial (up to deciding the ordering).