a linear transformation $A:{R^n} \to {R^m}$ with $m \times n$ matrix $[{a_{ij}}]$ Show that a linear transformation $A:{R^n} \to {R^m}$ with $m \times n$ matrix $[{a_{ij}}]$ can be written As 
$A = \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {{a_{ij}}} } {\varepsilon _i}{\pi _j}$.
I try to do it but I don’t understand ${\varepsilon _i},{\pi _j}$
Please help me. Thank
 A: I'm going to guess that
$$ \varepsilon_i = \begin{bmatrix}
0 \\
0 \\
\vdots \\
1 \\
\vdots \\
0
\end{bmatrix} \quad \text{ and } \quad \pi_j = [0,0 \dotsc, 1, \dotsc, 0],  $$
where $\varepsilon_i$ is an $m \times 1$ vector with a 1 in the $i^{th}$ component, and $\pi_j$ is a $1 \times n$ vector with a 1 in the $j^{th}$ component.  What happens when you multiply an $m \times 1$ vector with a $1 \times n$ vector?  You get an $m \times n$ matrix.  In particular, $\varepsilon_i \pi_j$ is an $m \times n$ matrix with a 1 in the $i^{th}$ row and the $j^{th}$ column and 0's everywhere else.  For example,
$$ \varepsilon_2 \pi_3 = \begin{bmatrix}
0 & 0 & 0 & \dotsb & 0 \\
0 & 0 & 1 & \dotsb & 0 \\
0 & 0 & 0 & \dotsb & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \dotsb & 0 \\
\end{bmatrix}. $$
Note that a linear transformation $A : \mathbb{R}^n \to \mathbb{R}^m$ is just a $m \times n$ matrix $(a_{ij})$.  Now
\begin{align*}
A &= a_{11} \begin{bmatrix}
1 & 0 & 0 & \dotsb & 0 \\
0 & 0 & 0 & \dotsb & 0 \\
0 & 0 & 0 & \dotsb & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \dotsb & 0 \\
\end{bmatrix} + a_{12} \begin{bmatrix}
0 & 1 & 0 & \dotsb & 0 \\
0 & 0 & 0 & \dotsb & 0 \\
0 & 0 & 0 & \dotsb & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \dotsb & 0 \\
\end{bmatrix} + \dotsb \\
&= a_{11} \varepsilon_1 \pi_1 + a_{12} \varepsilon_1 \pi_2 + \dotsb \\
&= \sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij} \varepsilon_i \pi_j
\end{align*}
A: The $\pi_j$ denotes the dual basis of orthonormal basis $\{\varepsilon_1,\varepsilon_2,...\varepsilon_n\}$ in $R^n$, $\pi_j(\varepsilon_i)=\delta_i^j$ ,    $\varepsilon_i$  can also represents one of the orthonormal basis in $R^m$.
