The space of linear maps So I have some lecture that I did not really understand and therefore I need a real example.
Let $V$ be a vector space. ${v_{1},...,v_{n}}$ is a basis for $V$
Let $W$ be a vector space. ${w_{1},...,w_{m}}$ is a basis for $W$
I want to build a basis for $Hom(V,W)$
So I need to build a linear maps such that $T_{ij}(v_{j})=w_{i}$
and $\forall k\neq j$    
$T_{ij}(v_{k})=0_{W}$   
But I do not really understand so I ask you to help me understand using an example.
Let $V=R^{2}$ 
and let $W=R^{3}$
both with the standard basis.
So how do I build the basis for $Hom(R^{2},R^{3})$ 
Thanks in advances !!
 A: For simplicity take the standard basis for $R^2$ (say $e_1,e_2$) and the standard basis for $R^3$ (say $f_1,f_2,f_3$). Then your construction says you have maps $T_{ij}$, where $i$ ranges from $1$ to $3$ and $j$ ranges from $1$ to $2$.
So for example $T_{11}$ is given by $T_{11}(e_1)=f_1$ and $T_{11}(e_2)=0$, So for an arbitrary element of $R^2$ you have $T_{11}(\begin{pmatrix}a\\b\end{pmatrix})=\begin{pmatrix}a\\0\\0\end{pmatrix}$. If you write that in matrix form you would get the matrix $\begin{pmatrix}1&0\\0&0\\0&0\end{pmatrix}$.
Now let's try another example $T_{32}$ is given by $T_{32}(e_1)=0$ and $T_{32}(e_2)=f_3$ by definition. Using linearity we again get $T_{32}(\begin{pmatrix}a\\b\end{pmatrix})=\begin{pmatrix}0\\0\\b\end{pmatrix}$. In matrix form this is $\begin{pmatrix}0&0\\0&0\\0&1\end{pmatrix}$.
In general, $T_{ij}$ written in matrix form (corresponding to the chosen bases) is given by the matrix with $1$ in row $i$ and column $j$ and $0$ on all other entries.
So, how do you get an arbitrary map in that basis: Let $T$ be arbitrary. Then you can compute $T(e_1)$ and $T(e_2)$ and express it in the basis $f_1,f_2,f_3$, i.e. $T(e_1)=t_{11}f_1+t_{21}f_2+t_{31}f_3$ and $T(e_2)=t_{12}f_1+t_{22}f_2+t_{32}f_3$ with $t_{ij}\in R$. Then $T$ is expressed in the basis given by the $T_{ij}$ as $T=\sum_{i,j}t_{ij}T_{ij}$. It's a good exercise to check that by applying both sides to the basis elements.
