Some basic question about vector spaces

[karseme]

I need some help understanding one task. I know that for some structure to be a vector space all axioms should apply. So if any of those axioms fails then the given structure is not a vector space. Anyway, I have a task where I need to check if \(\displaystyle \mathbb{C}^n_\mathbb{R} \) is a vector space. But, I am having trouble with understanding what \(\displaystyle \mathbb{C}^n_\mathbb{R} \) means. What is a complex vector space? I know that every vector space has 'V', which is a collection of 'vectors', and 'F' some field(real or complex), also two operations are defined with the given axioms being vector addition and scalar multiplication. But, where do \(\displaystyle \mathbb{C}\) and \(\displaystyle \mathbb{R} \) fit in this context? Is it not a symbol for complex vector space over a field \(\displaystyle \mathbb{R} \)? But, I am not sure how those relate to axioms. For, example \(\displaystyle \alpha (\beta a)=(\alpha \beta)a, \forall \alpha, \beta \in \mathbb{F}, \forall a \in V \)...and if we consider this task that I have then \(\displaystyle \alpha , \beta \in \mathbb{R} \). But, what about \(\displaystyle a \in ? \)? How does this relate to \(\displaystyle \mathbb{C}\)?

I searched for some definitions of a vector space even on the internet, but what confuses me also is what is connection between vector space and vectors? For example , \(\displaystyle \mathbb{R}^n \) is a vector space, then for n=1 \(\displaystyle \mathbb{R} \) is also a vector space. But, I don't see vectors anywhere if I have \(\displaystyle \mathbb{R} \), those are just real numbers. So, this term of vector space is kind of really abstract to me.

I would be grateful if someone could explain me this just a little bit. In a few sentences. What I would like is for somebody to make the meaning of a term 'vector space' a little bit clearer to me.

 

[Evgeny.Makarov]

Hi, and welcome to the forum!

Unfortunately, I don't recognize the notation $\mathbb{C}^n_{\mathbb{R}}$. It should be looked up in the source where it is encountered. Perhaps it is the space $\mathbb{C}\times\dots\times\mathbb{C}$ ($n$ times) over $\mathbb{R}$.

For, example \(\displaystyle \alpha (\beta a)=(\alpha \beta)a, \forall \alpha, \beta \in \mathbb{F}, \forall a \in V \)...and if we consider this task that I have then \(\displaystyle \alpha , \beta \in \mathbb{R} \). But, what about \(\displaystyle a \in ? \)?

If we consider $\mathbb{C}$ as a vector space over $\mathbb{R}$, then you are right that in your example $\alpha,\beta\in\mathbb{R}$ and $a\in\mathbb{C}$.

\(\displaystyle \mathbb{R}^n \) is a vector space, then for n=1 \(\displaystyle \mathbb{R} \) is also a vector space. But, I don't see vectors anywhere if I have \(\displaystyle \mathbb{R} \), those are just real numbers.

In linear algebra, the term "vector" does not mean anything similar to a pointed segment. It simply means an element of a vector space. So in this case ($\mathbb{R}$ as a vector space over $\mathbb{R}$), real numbers are both vectors and scalars, i.e., elements of the underlying field. In other vector spaces vectors can be functions and other abstract entities that don't have a geometric interpretation. You are right that the concept of a vector space is abstract, but this makes reasoning about vector spaces very general and applicable to many structures.

 

[karseme]

Let's say I kind of understand what you have explained. The only definition that I knew for vectors until now was actually that they are line segments that have starting point and ending point, and the properties which characterise them are their modulus(length), direction and orientation. So, basically you are saying that "vectors" in this, let's say, theory about vector spaces are not those pointed line segments. But, why would they call them "vectors" then, and "vector" space. Is there any reason that they are called the same name?

Thanks anyway, you did make it a little bit clearer. Hopefully, with time, it will become easier to understand all this.

 

[HallsofIvy]

Let's say I kind of understand what you have explained. The only definition that I knew for vectors until now was actually that they are line segments that have starting point and ending point, and the properties which characterise them are their modulus(length), direction and orientation. So, basically you are saying that "vectors" in this, let's say, theory about vector spaces are not those pointed line segments. But, why would they call them "vectors" then, and "vector" space. Is there any reason that they are called the same name?

Thanks anyway, you did make it a little bit clearer. Hopefully, with time, it will become easier to understand all this.

You said before that "I know that for some structure to be a vector space all axioms should apply" so you know that vectors, in this sense (which is really a physics concept, not mathematics", are not all possible vector spaces. The word "vector" is used in mathematics because those "physics" vectors do form a "vector space" in the mathematical sense. The "mathematics" vector is a generalization of the "physics" vector.