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The set of all polynomials is a vector space.

Created over 8 years ago, updated 10 days ago

Theorem: The set of all polynomials with coefficients in a field (\mathbb{F}), denoted (P) or (\mathbb{F}[x]), is a vector space over (\mathbb{F}).

  • Addition: ((a_0 + a_1x + \cdots)(b_0 + b_1x + \cdots) = (a_0+b_0) + (a_1+b_1)x + \cdots)
  • Scalar multiplication: (c(a_0 + a_1x + a_2x^2 + \cdots) = (ca_0) + (ca_1)x + (ca_2)x^2 + \cdots)

Zero vector: The zero polynomial (p(x) = 0).

This space is infinite-dimensional. The set ({1, x, x^2, x^3, \ldots}) is an infinite basis.

Note: Polynomials must have finitely many nonzero terms (each individual polynomial has finite degree), but there is no upper bound on the degree across the whole space.