Can someone please tell me what is the difference between $T:F^n \rightarrow F^k$ and $T:\mathbb{R}^n \rightarrow \mathbb{R}^k$, I know that are both made to define a Linear Transformation but I never understood what is the $F$ stands for.
Thank you in advance.
$\mathbb{R}$ refers to real numbers, but $F$ is a general field.
Therefore, $T:\mathbb{R}^n\rightarrow \mathbb{R}^k$ is a special case of $T:F^n\rightarrow F^k$, where $T$ transforms vectors from the vector space $\mathbb{R}^n$ to the vectors from the vector space $\mathbb{R}^k$.
In general, any set of object that have the following properties form a field:
0- Operations ($+$) (addition) and ($*$) (multiplication) are defined on the objects.
For $x,y,z\in F$, the following properties should hold:
1- Closure under addition and multiplication: $x+y, x*y\in F$
2- Associativity of addition and multiplication: $x+(y+z)=(x+y)+z$ and $x*(y*z)=(x*y)*z$
3- Commutativity of addition and multiplication: $x+y=y+x$ and $x*y=y*x$
4- Existence of additive and multiplicative identity elements: The should exist $0,1\in F$ such that $0+x=x$ and $1*x=x$
5- Existence of additive inverses and multiplicative inverses
6- Distributivity of multiplication over addition
Read more about fields here.
I guess $F$ is an arbitrary field. The real numbers $F=\mathbb{R}$ is just one example of a field.
Other fields you know, are $F=\mathbb{Q}$ and $F=\mathbb{C}$. But there are also other fields, for example finite fields (fields with only finitely many "scalars" in them).
