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I am having a difficult time proving that the set of all polynomials of degree less than n with coefficients in a field K where K is the real numbers is isomorphic with the set of all functions with values in K defined on a set S where S consists of n distinct points of R.

This a problem in Peter Lax's Linear Algebra book which I am reading.

The concepts I am finding difficult about this problem is how to tell if these linear spaces are onto and one-to-one with each other. So if you can explain that a bit in-depth and step by step that would be great.

Thanks in advance for any help!

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  • $\begingroup$ Vector spaces aren't onto or one-to-one- you want to give a linear transformation from one vector space to the other is onto and one-to-one. Your first step should be to find some linear transformation. Say for example that $n=3$, and $p(x)=2x^2-3x+4$. Now say that your other vector space is the set of $K$-valued functions defined on the set of points $S=\{a_1, a_2, a_3\}$. Can you choose some such function $f$ that naturally corresponds to our polynomial $p$? If you can, you can generalize that to a map between your vector spaces, and then prove that that map is linear, onto, and one-to-one. $\endgroup$ – Kevin Long Oct 15 at 19:05
  • $\begingroup$ And here, $K$ apparently means $\mathbb{R}$. $\endgroup$ – Kevin Long Oct 15 at 19:06

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