# Vector spaces in terms of polynomials?

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.

• 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!
– lulu
Apr 15 '20 at 14:04
• 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. Apr 15 '20 at 14:05
• 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. Apr 15 '20 at 14:09
• 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) Apr 15 '20 at 14:30
• That should read a mathematical "vector space", not vector Apr 15 '20 at 16:16

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 $$k \cdot p(x)$$.