Explanation of a question

Solution 1:

$\delta_x$ is the indicator function of $x \in M$, i.e. $\delta_x(x) = 1$ and $\delta_x(y) = 0$ if $y\neq x$.

$\{\delta_x | x \in M\}$ is the set of all indicator functions of the items in $M$, so $[\{\delta_x | x \in M\}]$ is their span.

So we need to prove: given a finite set $M$, the linear maps from $M$ to some field $K$ is spanned by the set of indicator functions.

Proof: We only prove $\text{Map}(M,K) \subset [\{\delta_x | x \in M\}]$ and the other direction follows from definition.

Consider any $f \in \text{Map}(M,K)$. For $m_1, m_2,... m_n \in M$, we have

$$f(m_k) = v_k$$

for some $v_k\in K$. Now

$$f(x) = \sum_k v_k \delta_{m_k}(x)$$, therefore

$$f = \sum_k v_k \delta_{m_k} \in [\{\delta_x | x \in M\}]$$.