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.

 

[Ishuda]

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.

In the sense given a would just be an n-tuple of complex numbers
a = (c₁, c₂, c₃, ...,c_n)
Addition would generally be defined component wise, i.e.
a₁ + a₂ = (c₁₁, c₂₁, c₃₁, ...,c_n1) + (c₁₂, c₂₂, c₃₂, ...,c_n2) = (c₁₁+c₁₂, c₂₁+c₂₂, c₃₁+c₃₂, ...,c_n1+c_n2)
etc.

Thus the vectors can be considered as just n-tuples. In general the elements of the vector can come from an abelian (commutative) group (calling the operator addition) and the scalers need to be a field in which multiplication by a member of the field with a component of the n-tuple 'makes sense'.

As an example take as our set G the n-tuples whose element consist of members from the abelian group the integers modulo 3, i.e.
a = (g₁, g₂, g₃, ..., g_n)
where each of the g's is either 0, 1, or, 2. Certainly, if we define additive operations component wise with the usual modulo arithmetic, the set G will satisfy the commutativity and associativity properties, there is an Additive identity and all elements have an additive inverse. Thus
(0, 1, 2) + (1, 2, 0) = (0+1, 1+2, 2+0) = (1, 0, 2)
(1, 2, 0) + (2, 1, 0) = (1+2, 2+1, 0+0) = (0, 0, 0)
We can take as our field the elements from the field the integers modulo 5, i.e. 0, 1, 2, 3, and 4 and use usual modulo multiplication, i.e.
4 (0, 1, 2) = (4*0, 4*1, 4*2) = (0, 1, 2)
2 (0, 1, 2) = (2*0, 2*1, 2*2) = (0, 2, 1)