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## Homework Statement

a) Let ##D_n(\mathbb{k}) = \{A \in M_n(\mathbb{k}) : a_{ij} = 0 \iff i \neq j\}##

Prove that ## D_n(\mathbb{k}) \cong \mathbb{k}^n ##

b) Prove that ##\mathbb{k}[X]_n \cong \mathbb{k}^{n+1}##

I have one other exercise, but I would like to resolve it on my own. However, an unfamiliar bit of notation appears that I would like clarification on: what is ##\mathbb{k}^{\{x\}}##?

**The attempt at a solution**

I am unfamiliar with the notation used to prove isomorphism between two mathematical objects. I can see, for example, that the group of the diagonal matrices is a vector space, as is ##\mathbb{k}^n##, and that therefore there should exist a linear transformation that maps the elements of one to the other.

Would this look something like:

##\phi : D_n(\mathbb{k}) \rightarrow \mathbb{k}^n...## but what would the map of this linear transformation look like? Am I mapping ##\{A \in M_n(\mathbb{k}) : a_{ij} = 0 \iff i \neq j\}## to ##\{x, y, \ldots , n\} \in \mathbb{k}^n## or something similar?

Thanks for any help.