Vectorspace set as dimension I encountered some notation in my mathematics exercises which I couldn't make sense of and couldn't find on the internet.
Usually, a vector space is written like this: $K^n$. For example, $\mathbb{R}^3$ tells me that my vector space consists of vectors with three entries, each being a real number.
The notation I encountered is this: $F_2^M$. I know what $F_2$ is (only 1 and 0 are in $F_2$). But M is given to be a set. So this cannot mean that I have vectors with M entries each being in $F_2$.
So can anyone tell me what this notation means? Thank you!
 A: It might mean the set of all functions $M \to F_2$, which does have a vector space structure: $(f+g)(m) = f(m) + g(m)$ and $(\lambda f)(m) = \lambda f(m)$.
A: Maybe it means thar its the set of vectors from different lengths?
Eg  $F_2^{1,3}$ would contain 0, 1, [0 0 0], [0 0 1] ....
A: I would think that this vector space is the power set of $M$ (often denoted $2^M$).  Vector addition is symmetric difference.  The zero-vector is the empty set.   Scalar multiplication is given by $1\cdot A=A$ and $0\cdot A=\emptyset$ 
A: This notation is often used to mean that you have $|M|$ as the dimension of your vector space over $F$, and that the set $M$ is in fact taken as a basis.
For example, if you have $\mathcal{V}$ as the set of vertices in a graph, then you might want to consider $\mathbb{Q}^{\mathcal{V}}$ to be vectors with an entry for each element of $\mathcal{V}$. This notation is convenient because we do not need to explicitly specify an ordering for the set, it is just a given that each entry of the vector corresponds to an element. This gives a nice way, in particular, to discuss eigenvectors of a graph.
(Note that this idea corresponds to thinking of elements of $F^{M}$ as functions from $M \to F$: a vector has a coordinate in $F$ corresponding to each element of $M$.)
