Why Do Hermite Polynomials Form a Basis for P3?

[Moonshine]

A question in my textbook asks me to show that the first four Hermite polynomials form a basis of P3.

I know how to do the problem, but I don't really understand what is behind the scenes.

Why can we use the coefficients of a polynomial as a column vector? I don't really know how to ask the question (that means I'm probably really lost, haha).

Also, how do you say P3 (like if I was speaking to someone).

What exactly is polynomial space? Can it be visualized? I'm confused.

 

[Cantab Morgan]

I think you're thinking of vector spaces exclusively as sets of columns of numbers. But vectors don't have to be columns of numbers. Vector spaces are sets whose elements behave by certain rules. For example, there's a binary operation on those elements (addition) that carries pairs of elements to a third element. This operation is associative, commutative, and so forth.

Any set that obeys all those rules is a linear space (or vector space, or space). It doesn't have to be a column of numbers. So, if you define addition of polynomials in the expected way, then it turns out that polynomials of a certain degree (or less) form a vector space.

To pick a basis is to choose an invertible function from the vector space of polynomials onto the "ordinary" vector space of tuples of real numbers. When you say, "Why can we use the coefficients of a polynomial as a column vector?" What you're really doing is choosing one mapping among many, carrying polynomials into R^n.

The Hermite polynomials form another choice of basis. They represent a different mapping from the space of polynomials to R^n. However, they have the advantage of being orthogonal, given the choice of a certain inner product.