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".
Best Answer
A vector space over a field simply means it's a vector space where your scalars (used in scalar multiplication) come from a field. Usually when we talk about a vector space, we mean vector space over a field. We can consider vector spaces over other rings such as the integers, but these are usually called modules.
An algebra over a field is a vector space along with a multiplication of your vectors. For this multiplication to be useful, you want it to be bilinear, that is if we multiply $$(a_{1}u_{1} + b_{1}v_{1})*(a_{2}u_{2}+b_{2}v_{2}),$$ we want this to be equal to $$a_{1}a_{2}(u_{1}*u_{2})+a_{1}b_{2}(u_{1}*v_{2})+b_{1}a_{2}(v_{1}*u_{2})+b_{1}b_{2}(v_{1}*v_{2}).$$ (here $a_{1}, b_{1}, a_{2}, b_{2}$ are scalars in my field, and $u_{1},v_{1}, u_{2}, v_{2}$ are vectors).