What is meant by "Maps" and |.| and "canonical basis" in abstract algebra (in the given context)?

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 of $\{1,2,3\}$ and $\{1,2\}$, or no?

Any suggestion regarding how to approach this question is appreciated.

Answer 1

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.

Answer 2

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.

Answer 3

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 $\text{Maps}$.