\\section{Matrices as Functions}

 Another way to think of a matrix is as a \'mapping\' from vectors to vectors. This can be thought of an actual transform (generally, a scaling or rotation) or as a change in coordinate system.
    
   For example, the following matrix \'projects\' a two-dimensional plane onto the line $y = -x$: 

$$ [
\\begin{array}
1&-1\\\\
-1&1
\\end{array}] \\CircleTimes [
\\begin{array}
x\\\\
y
\\end{array}] =

[\\begin{array}
x-y\\\\
y-x
\\end{array}] $$
  
   A very important point is that information is being lost in this transform. For example, the point $(2,2)$ is the \'image\' of both $(5,3)$ and $(12,10)$ in the original space. The fact that this mapping \'loses\' a dimension is reflected in the fact that the two rows of the matrix are (trivially) linearly dependant. This matrix has \'rank\' ONE, since it can only transform ONE dimension without losing information.

Note that a \\it{rectangular} matrix of size $m \\times n, m > n$ can have at most rank n.