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Can someone explain what $\operatorname{End}_{\mathbb{C}}(\mathbb{C}[x])$ is? I just want to know what its elements look like.

In the definition, it says that for a field $K$ and $n \in \mathbb{N}$, $M_{n}(K)$ is an algebra over $K.$

I understand what an algebra over $K$ is now.

$M_n(K) = \operatorname{End}_{\mathbb{C}}(\mathbb{C}[x])$ where $M_n(K)$ is the set of all linear transformations from $K^n$ to itself.

Like what is a typical element of $\operatorname{End}_{\mathbb{C}}(\mathbb{C}[x])$? Is it a matrix multiplied by some complex number?

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    $M_n\le$f$t(K\right) = \mathrm{End}_{\mathbb C}\le$f$t(\mathbb C\left[x\right]\right)$? How is that supposed to be true?2011-11-28

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As a set $\operatorname{End}_{\mathbb{C}}(\mathbb{C}[x])$ is the set of endomorphisms with domain and codomain $\mathbb{C}[x]$. In your instance it's algebra endomorphisms. So, it's functions $f:\mathbb{C}[x]\rightarrow\mathbb{C}[x]$ that preserve the ring structure on $\mathbb{C}[x]$ as well as the structure of $\mathbb{C}[x]$ as a vector space over $\mathbb{C}$.

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    I often see things like $\operatorname{End}_{\mathbf{C}-\text{alg}}(\mathbf{C}[x])$, which is admittedly a pain to write down every time.2011-11-30
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You could think like $M_\infty(\mathbb{C})$, where you matrix are indexed in the natural and add just one of the follow conditions the rows or the columns entries are almost all zero, that is only a finite number of entries are distinct from zero. Consider that $\mathbb{C}[x]$ is a vector space with basis $\{X^n\}_{n\in \mathbb{N}}$ and use the universal property of the basis.

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    You have to add only one of the conditions, it depends of how you define the coordinates, as a row or as column.And also dependes how notation you use for functions f(x) or (x)f.2011-11-28