Understanding an Algebra over a Field

Question

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:

1. What it means to be a vector space over a field. Wondering if it's different than just a vector space.
2. 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.
3. 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".

Answer 1

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).

Answer 2

1. A vector space over a field $\mathbb{F}$ is just one level more of abstraction. $V = \mathbb{R}^n$ is a vector space over the field $\mathbb{R}$. A vector space $V = \mathbb{C}^n$ is a vector space over the field $\mathbb{C}$. We could have other fields like finite fields.
2. A bilinear product is a mapping involving three vector spaces. Think of matrix multiplication. A matrix $A \in \mathbb{R}^{n \times m}$ is a vector in the vector space $V=\mathbb{R}^{n \times m}$. A similar statement holds for another matrix $B \in \mathbb{R}^{m \times p}$. Multiplying them together, $AB$ together yields another matrix for which again a similar statement holds. In all this the field is $\mathbb{R}$.
3. No. An algebra over a field is defined to be exactly the structure you mention; its one vector space $V$ equipped with a bilinear product operation.

Answer 3

1. You wrote that you are “somewhat familiar with vector spaces”. In those vector spaces you can multiply vectors by scalars. Where do they come from? Are they real numbers? Then you have a vector space over the field $\mathbb R$ of real numbers. Are they complex numbers? Then you have a vector space over the field $\mathbb C$ of complex numbers. And so on.
2. A bilinear product is not “a set of 3 vector spaces over the same field”. It's a map from $V_1\times V_2$ into $W$ with certain properties, in which $V_1$, $V_2$ and $W$ are vector spaces over some field. In the case of an algebra, $V_1=V_2=W$.
3. No. It's a particular case of an algebraic structure.