TITLE: Endomorphisms of a finite dimensional vector space
QUESTION [6 upvotes]: From Humphreys' Introduction to Lie Algebras and Representation Theory:

If $V$ is a finite dimensional vector space over $F$, denote by $\text{End }V$ the set of linear transformations $V\rightarrow V$. As a vector space over $F$, $\text{End }V$ has dimension $n^2$ ($n=\text{dim }V$), and $\text{End }V$ is a ring relative to the usual product operation.

1) Why does $\text{End }V$ have dimension $n^2$? I think of an example like $V=\mathbb{R}^n$, with the basis $e_1=(1,0,\ldots,0),e_2=(0,1,\ldots,0),\ldots,e_n=(0,0,\ldots,1)$. Then a linear transformation $V\rightarrow V$ can take $e_1$ to $e_j$ for any $j=1,2,\ldots,n$, take $e_2$ to $e_j$ for any $j=1,2,\ldots,n$, and so on. This should give rise to $n^n$ independent linear transformations.
2) "$\text{End }V$ is a ring relative to the usual product operation." What is the product operation referred to here?

REPLY [4 votes]: 1.
Hint
Let $f_{p,q}:V\to V$ defined by
$$f_{p,q}(e_i)=\delta_{pi}e_q$$
Prove that $(f_{p,q})_{1\leq p,q\leq n}$ is a basis for $\mathrm{End}(V)$. Conclude.
2.The product operation  is the  composition of the endomorphisms