# Isomorphism of Linear space of polynomials with degree less than n and $V_n(R)$

I need to show that there is an Isomorphism of Linear space of polynomials with real coefficients with degree less than n and $V_n(R)$. Now $V_n(R)$ is the linear space where the vectors are n-tuples of real numbers and the scalars are real numbers with addition and scalar multiplication defined by $$(\alpha^1,....,\alpha^n)+(\beta^1,....,\beta^n)=(\alpha^1+\beta^1,....,\alpha^n+\beta^n)$$ and $$\gamma(\alpha^1,....,\alpha^n)=(\gamma\alpha^1,....,\gamma\alpha^n)$$

So, I need to either find a linear transformation from $V_n(R)$ to the set of polynomials of degree less than n, or show that their dimensions are the same. Which one is easier to show? How do I start? I am asking these questions to myself first of all and I appreciate all your input.

$T(P_{n-1})=(\alpha^1,....,\alpha^n)$ seems like the linear transformation I am looking for, right? I know both of these spaces have degree n, but how does one show that?

Let $P_h$ and $P_m$ be two polynomials of power less than n-1.

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## 1 Answer

both approaches are equally easy I would say, at least in this particular case. Also, the time it took you to formulate this question pondering which approach would work is probably more than the time it would have taken you to just solve the problem using any approach (or even both). It is a common mistake in the first steps in mathematics to spend enormous amounts of time pondering which approach to try. Instead, just try. Intuition will come later with experience.

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