Sketching vector diagram I need to sketch a diagram which shows what happens when a 2x2 matrix is applied to a lattice. What would be the best software to perform such a thing?

I've used MATLAB to plot such transformations.

I am assuming you have a lattice of 2D points $x$ and your matrix $A: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. 
So you'll get another set of 2D points $x'$ where $x' = Ax$. I use MATLAB to plot both sets of points.
E.g. $A$ is the 2x2 rotation matrix defined below with $\theta = \pi/4$.
$ A = 
\begin{bmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix}
$
The MATLAB code:
theta = pi/4;
A = [cos(theta) -sin(theta); sin(theta) cos(theta)];
x = [1 1 1 0 0 0 -1 -1 -1;-1 0 1 -1 0 1 -1 0 1];
xp = A*x;

The new display code:
v = xp - x;
axis([-2 2 -2 2]);
hold on;
plot(x(1,:),x(2,:),'*k');
plot(xp(1,:),xp(2,:),'*r');
quiver(x(1,:),x(2,:),v(1,:),v(2,:),0)

And the diagram:

