Question: I found this abstract algebra question in a previous test paper:

> 
> Suppose $\Bbb{F}$ is a field and $\mathrm{X}$ is a non-empty set. Then
>  $\text{Maps}(\mathrm{X},\Bbb{F})$ is a vector space over $\Bbb{F}$. 
> 
> 
> * If $|\mathrm{X}|=3$ then find a canonical basis of $\text{Maps}(\mathrm{X},\Bbb{F})$.
> * Can you recognize $\text{Maps}(\mathrm{X},\Bbb{F})$ where $\mathrm{X}=\{1,2,3\}\times \{1,2\}$ & $\Bbb{F}=\Bbb{R}$ ?
> 
> 
> 

I'm not sure what $\text{Maps}$ mean here? Is it a standard term in abstract algebra? Also, what exactly is meant by "canonical" basis? Moreoever, what is meant by $|\mathrm{X}|$? Does it refer to the number of elements in the set $\mathrm{X} $?

For the second case is $\mathrm{X}$ just the [Cartesian Product](https://en.wikipedia.org/wiki/Cartesian_product) of $\{1,2,3\}$ and $\{1,2\}$, or no?

Any suggestion regarding how to approach this question is appreciated.

Answer: 
The notation $\operatorname{Maps}(X,\mathbb{F})$ is not “standard”, but it can be easily understood: the set consists of all maps $X\to\mathbb{F}$.

You can endow it with pointwise addition, $f+g\colon x\mapsto f(x)+g(x)$, and pointwise scalar multiplication, $af\colon x\mapsto af(x)$.

A “canonical basis” (the name is not well chosen, in my opinion) consists of the maps that have value $1$ at a single point of $X$ and $0$ on all others.

A map $\{1,2,3\}\times\{1,2\}\to\mathbb{F}$ associates to each pair $(i,j)$ (with $1\le i\le 3$ and $1\le j\le 2$) an element of $\mathbb{F}$: does this ring a bell?

> 
>  Matrices! This is an abstract definition for them.
> 
> 
>
Yes, $|X|$ means the number of elements of $X$. And "maps" means "functions". The basis is "canonical" because it is so obvious and natural. 

Also $\times$ in this context denotes the Cartesian product.