Visual representation of matrices I am used to seeing most basic mathematical objects being visually represented (for instance, a curve in the plane divided by the xy axis; the same goes for complex numbers, vectors, and so on....), However, I never saw a visual representation of a matrix. I do not mean the disposition in rows and columns, of course. 
I would like to know whether they can be graphically represented in some intuitive way. If so, how? Could you illustrate it, say, with a 2x2 quadratic matrix?
Thanks a lot in advance

You can represent a $2 \times 2$ matrices $A = \left[\begin{smallmatrix}a & b\\c & d\end{smallmatrix}\right]$ as a parallelogram $Q_A \subset \mathbb{R}^2$ with
vertices $(0,0), (a,c), (b,d)$ and $(a+b,c+d)$. If one identify the plane $\mathbb{R}^2$ with $M^{2\times 1}(\mathbb{R})$, the space of $2 \times 1$ column matrices, then $Q_A$
is the image of the unit square $[0,1] \times [0,1]$ under linear transform
$$[0,1] \times [0,1] \ni \begin{bmatrix}x \\ y\end{bmatrix}
\quad \mapsto \quad \begin{bmatrix}x' \\ y'\end{bmatrix}
= \begin{bmatrix}a & b\\c & d\end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix} \in Q_A
$$
Since a linear transformation is uniquely determined by its action on a basis,
this provides a faithful represent of the $2 \times 2$ matrices.
Under this representation, some geometry related operations now correspond to familiar geometric shapes. e.g.


*

*The matrix $\left[\begin{smallmatrix}s & 0\\0 & s\end{smallmatrix}\right]$ represents a scaling of geometric objects. It corresponds to a square of side length $s$, axis aligned with the standard $x$- and $y$-axis.

*The matrix $\left[\begin{smallmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \;\cos\theta\end{smallmatrix}\right]$ represents a counterclockwise rotation of angle $\theta$. It corresponds to the unit square rotated counterclockwisely for angle $\theta$. 

*The matrices  $\left[\begin{smallmatrix}1 & m\\ 0& 1\end{smallmatrix}\right]$
and           $\left[\begin{smallmatrix}1 & 0\\ m& 1\end{smallmatrix}\right]$
represent sheer mappings in horizontal and vertical directions. They can visualized as parallelogram with one pair of its sides staying horizontal and vertical respectively.
This sort of shapes provides a useful visual mnemonics for what are the effects of those matrices (when viewed as a transformation of the plane).
Finally, one can also use this to introduce the concept of determinant to students.


*

*What is the the determinant of a matrix $A$? It is just the area of $Q_A$.

*What does it mean $\det A < 0$? It just mean the $Q_A$ has been flipped.


I'm sure there are other ways to use this visualization as a teaching tool.
