dimension of that vector space? I would like to know how to prove that the dimension  of $L(E,E)$ (that is the set of linear maps from  $E\rightarrow E$) where $E$ is a finite dimensional vector space ($dim E=n$). I know that the dimension is $n^2$ but I do not know how to prove it, thanks.
 A: let us consider a basis $e_i$, $i=1\ldots n$ of $E$, and an element $A\in L(E,E)$.
$A$ is caracterized via $A(e_1)\ldots A(e_n)$ who are elements of
$E$, and so
$$
A(e_i) = \sum_{j=1}^n A_{i,j}e_j
$$
So the elements $E_{i,j}\in L(E,E)$ defined by
$$\begin{array}{lll}
E_{i,j} (e_k)&=& e_j \text{  if }i=k\\
&=&0\text{ otherwise}\end{array}$$
is a family of cardinal $n^2$ and who generates $L(E,E)$.
Hence $\dim L(E,E)\leq n^2<\infty$.
A: Choosing a basis for $E$, we can represent operators in $L(E, E)$ by matrices in this basis.  Look at the matrices $E_{mn}$ such that $E_{mn}$ has the entry $1$ in the $m-n$ position and zeroes everywhere else.  They are linearly independent since $\sum \alpha_{ij} E_{ij}$ is the matrix $[\alpha_{ij}]$; it is zero if and only if $\alpha_{ij} = 0$ for all $i, j$.  This argument also shows how any matrix may be expanded in terms of the $E_{mn}$.  Thus the $E_{mn}$ form a basis for $L(E, E)$.  There are $n^2$ such $E_{mn}$; thus $\dim L(E, E) = n^2$. 
Edit:  Note in response to comment of Dror:  If $A \in L(E, E)$ with basis $\mathbf e_i$ of $E$, then we have $A(\mathbf e_i) = \sum \alpha_{ij}\mathbf e_j$ for some matrix $[\alpha_{ij}]$, and any such matrix determines a linear operator in $L(E, E)$ since it defines $A(\mathbf e_i)$; indeed we have
$A(\sum_i w_i \mathbf e_i) = \sum_i w_i A(\mathbf e_i) = \sum_i w_i \sum_j \alpha_{ij} \mathbf e_j = \sum_j (\sum_i w_i \alpha_{ij}) \mathbf e_j; \tag{1}$
the matrix $\alpha_{ij}$ operates on vectors of components $(w_1, w_2, \ldots, w_n)$ on the right; the components $w_i$ form a "literal" vector; it's a duality thing.  And since the vectors $\mathbf e_i$ are given in abstracto, it matters little in what space they lie, just so it is of finite dimension.  Vectors, matrices, tensors, the basic idea is the same.  End of Note.
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
