# Linear algebra, proof that it's an isomorphism.

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.

Hint: Send $$a_0+a_1x+a_2x^2+\cdots+a_nx^n \in V$$ to $$(a_0,a_1,a_2,\ldots,a_n) \in K^{n+1}$$.
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).
• Any polinomial of degree less or equal than $n$ can be written as $a_0+a_1 x + a_2 x^2 + ... + a_n x^n$, where any of these a_i can be zero; now, make a vector out of these $n+1$ elements of $K$, and there goes your isomorphism. (No need to feel sorry! Math is hard.) – Simone Ramello Feb 9 at 21:52