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: 
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.
Since each $f \in \text{Maps}$ is determined by the values $f(a\_1),f(a\_2),f(a\_3)$ in the field, the natural basis for $\text{Maps}$

$$x\_i(a\_j)= \delta\_{ij}$$

As a quick check, for any $f \in \text{Maps}$ with $\alpha\_j=f(a\_j)$ look at 

$$q(t)=\sum\limits\_j \alpha\_jx\_j(t)$$ Then note that, $$q(a\_j)=\alpha\_j=f$$

It remains to show that the $x\_j$'s are linearly independent which then defines a basis for $\text{Maps}$.