Prove Left Module Over Ring of Matrices is Cyclic

Question

Let $R=\text{Mat}_{n\times n}(\mathbb{C})$ and $M=\mathbb{C}^n$ be the left $R$-module with $A\cdot v=Av$ (given by left matrix multiplication). Prove that $M$ is a cyclic $R$-module.

This question seems to be fairly straightforward but I have no idea how to go about proving it.

The definition of cyclic I am working with is $M$ is cyclic if $M=R\{a\}=\{ra|r\in R\}$ for some $a\in M$.

Edit (Proof)

Let $R=\text{Mat}_{n\times n}(\mathbb{C})$ and let $M=\mathbb{C}^n$ be a left $R$-module.

Consider the element $v=e_1=\begin{bmatrix}1\\0\\\vdots\\0\end{bmatrix}\in M$.

Then for any $A\in R$ where $A=\begin{bmatrix}c_{11} & ... & c_{1n}\\c_{21} & ... & c_{2n}\\ \vdots\\c_{n1} & ... & c_{nn}\end{bmatrix}$ for $c_{ii}\in\mathbb{C}$,

$A\cdot v=\begin{bmatrix}c_{11} & ... & c_{1n}\\c_{21} & ... & c_{2n}\\ \vdots\\c_{n1} & ... & c_{nn}\end{bmatrix}\cdot \begin{bmatrix}1\\0\\\vdots\\0\end{bmatrix}=\begin{bmatrix}c_{11}\\c_{21}\\\vdots\\c_{n1}\\\end{bmatrix}$.

If we consider $Rv=\{Av|A\in R\}=\left\{\begin{bmatrix}c_{11}\\\vdots\\ c_{n1}\\\end{bmatrix}|c_{11},...,c_{n1}\in\mathbb{C}\right\}$, it is clear that this set of vectors produces all of $\mathbb{C}^n$. Therefore, $M$ is a cyclic $R$-module.

Answer 1

Take any $v\in M\setminus 0$ whatsoever and show that $Rv=M$.

Do an example: take $n=2$, and, say $v=e_1$. What is the set of vectors you can get as $Av$ with $A$ a matrix?

After you have done that little example, think about what happens when $n=3$.

After that, you will see how to do this.

Answer 2

Hints.

• For every $v\in\mathbb{C}^n$ there exists a linear map $f\colon\mathbb{C}^n\to\mathbb{C}^n$ such that $f(e_1)=v$, where $\{e_1,\dots,e_n\}$ is the standard basis.

• What are the linear maps $\mathbb{C}^n\to\mathbb{C}^n$? (Look at their associated matrices with respect to the standard basis.)