How does one show that the set of polynomials is infinite-dimensional? Does one begin by assuming that a finite basis for it exists, and then reaching a contradiction?(adsbygoogle = window.adsbygoogle || []).push({});

Could someone check the following proof for me, which I just wrote up ?

We prove that V, the set of all polynomials over a field F is infinite-dimensional. To do so, assume on the contrary that it is finite-dimensional, having dimension n. Then there exists a basis for V having n elements.

Since the following set is linearly independent and has n elements, it is also a basis for V:

[itex] β = \{ 1, x, x^{2}...x^{n-1} \} [/itex]

Thus every polynomial is expressible as a linear combination of the vectors in this set.

But then [itex]x^{n} \in span(β) [/itex] which implies that [itex] β \cup \{x^{n}\} [/itex] is linearly independent. This is clearly false, hence a contradiction. Thus the vector space of polynomials is infinite dimensional.

Is it completely correct?

BiP

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# Set of polynomials is infinite dimensional

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