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 $(a_0,a_1,a_2,\ldots,a_n) \in K^{n+1}$.

  • $\begingroup$ Thank you very much! $\endgroup$ – Scoofjeer Feb 9 at 21:51

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).

  • $\begingroup$ I don't get it. I'm sorry. $\endgroup$ – Scoofjeer Feb 9 at 21:36
  • $\begingroup$ 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.) $\endgroup$ – Simone Ramello Feb 9 at 21:52
  • $\begingroup$ Indeed it is. Thanks for the explanation, is really helpful. $\endgroup$ – Scoofjeer Feb 9 at 21:55

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