TITLE: What's a good primer from linear algebra to spherical harmonics?
QUESTION [7 upvotes]: I need a topic, a primer, that will be able to introduce me to spherical harmonics and how to translate and use them with the usual tools of linear algebra and calculus, namely matrices, polynomials and derivatives for example .
In other words, I would like to know enough to handle and compute harmonics.
What topics do you suggest I should touch to get up and running with spherical harmonics starting with a linear algebra and calculus background ?

REPLY [3 votes]: You should read about these topics:

Vector spaces (I think you already know what this is)
Metric spaces (A vector space + a norm)
Hilbert spaces (A metric space with a finite or infinite number of dimensions)
Fourier transform (The projection of a vector belonging to a Hilbert space in an orthogonal basis)
Spherical coordinates (Really easy to understand, check Wikipedia)

Once you get that, you'll be able to understand the spherical harmonics:
An orthogonal basis for the Hilbert space of functions in spherical coordinates that only vary with $\theta$ and $\phi$ under the norm $L_2$.
P.S. The spherical harmonics are the sinus and cosinus of functions in 3D that don't change with $r$.
I'm not being very rigorous because I wanted to give you an eagle view on the subject.