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I am able to understand when we speak of vectors, what we mean by a vector space and its basis. A basis is the smallest spannable set of vectors that can span the vector space.

But what does it mean for us to talk about a vector space for polynomials? How can we define a basis for a polynomial? The way I think about it polynomials are not vector spaces. They are not arrows with directions and magnitudes.

I know my question is a little naive. But I did not know how else to express my confusion.

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  • $\begingroup$ You can add polynomials, there is a $0$ polynomial, you can multiply polynomials by constants and the distributive law holds...that's all you need to make a vector space! $\endgroup$
    – lulu
    Apr 15 '20 at 14:04
  • $\begingroup$ A vector space is only the abstractisation of the notion of vector: its elements can be added and multiplied by scalars (elements of a field), with the usual properties. Just like the colour ‘orange’ is the abstractisation of the colour of the eponymous fruit. $\endgroup$
    – Bernard
    Apr 15 '20 at 14:05
  • $\begingroup$ Nothing in the definition of a vector space refers to "arrows" or "directions and magnitudes". And regarding your question about a basis for the space of polynomials, the set $\{ 1,x,x^2,x^3,\ldots \}$ clearly forms a basis. $\endgroup$ Apr 15 '20 at 14:09
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    $\begingroup$ In the context, vectors are not necessarily "objects with magnitude and direction". Instead, a mathematical vector is any set of objects that can be added together and "scaled" (multiplied by a number) $\endgroup$ Apr 15 '20 at 14:30
  • $\begingroup$ That should read a mathematical "vector space", not vector $\endgroup$ Apr 15 '20 at 16:16
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In physics, vectors are often described as objects with "magnitude and direction". However, this description does not apply to every object that can be called a vector in the mathematical sense. Mathematically speaking, a vector space is any set $V$ of objects for which the notions of "addition" (finding $u+v$ for vectors $u,v \in V$) and scaling (finding $ku$ for any number $k$ and vector $v \in V$). We also require that addition and multiplication behave "naturally", which is to say that $V$ satisfies the vector space axoims.

With that said, the set $\mathbb P_n[x]$ of polynomials with degree at most $n$ fits this description. In particular: for two polynomials $p(x),q(x)$, and for any number $k$, we could sensibly compute $p(x) + q(x)$ or $k \cdot p(x)$.

Interestingly, vector spaces can be given an additional structure of "direction". In particular, defining an inner product allows us to consider the "angle" between two vectors. This idea is important in Harmonic analysis, where one often makes use of the fact that certain trigonometric polynomials are "orthogonal".

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