I am trying to understand the components of an algebra over a field.
an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.
I am familiar with the definition of a field, and sort of familiar with vector spaces. I am not familiar with bilinear products and the meaning of a "vector space over a field". Wikipedia says a bilinear map is a set of 3 vector spaces over the same field.
I am wondering the following:
- What it means to be a vector space over a field. Wondering if it's different than just a vector space.
- Since the 3 vector spaces are different, but they all "work on" the same underlying set, wondering how their operations are compatible. It would be helpful to see an example of a bilinear map/product as vector spaces over a field.
- If "an algebra" in this context is the same as the generic definition of an algebra as "a set plus a set of operations on that set".