I am reading through, Pressley's elementary differential geometry, and in chapter 6 on first fundamental forms, there is a concept of symmetric bilinear form, which I misunderstand.
Here is the text:
Suppose that $\sigma(u, v)$ is a surface patch of $S$. Then, any tangent vector to $S$ at a point $p$ in the image of $\sigma$ can be expressed uniquely as a linear combination of $\sigma_u$ and $\sigma_v$. Define maps $du : T_pS \rightarrow R$ and $dv : T_pS \rightarrow R$ by $du(v) = \lambda, du(v) = \mu$ if $ v = \lambda \sigma_u + \mu \sigma_v $. for some $\lambda, \mu \in R$. It is easy to see that $du$ and $dv$ are linear maps. Then, using the fact that $< , >$ is a symmetric bilinear form, we have: $<v,v> = \lambda^2<\sigma_u,\sigma_u> + 2\lambda\mu<\sigma_u,\sigma_v> + \mu^2<\sigma_v, \sigma_v> $.
Could someone explain this concept? I found a definition of symmetric bilinear form on wikipedia, but did not find it enlightening. (i mainly do not understand where the last equality comes from but do see how dot product satisfies the definition :https://en.wikipedia.org/wiki/Symmetric_bilinear_form)