The main importance, IMHO, is that Linear Algebra is a language, it gives us the descriptive power to talk about things that would otherwise be either very cumbersome or very vague.
For instance, one of the things matrices give us is the power to talk about a system of equations as a single unit, and do a variety of operations that make sense to do on the system as a whole.
Another thing it gives us is the power to speak rigorously about some geometric concepts. For example, consider the surface given by the parametric equations:
x(s, t) = s
y(s, t) = s^2 * t
with -1 < s < 1 and -1 < t < 1
If you plot this suface, you will notice that it's kinda sort of pinched into a 1-D surface at the origin. We can prove this rigorously as follows:
First, compute the jacobian of the above equations. The jacobian of a system of functions is simply the matrix who's i-th row and j-th column contains the derivative of the i-th function with respect to the j-th variable. In this case, the jacobian of the transformation is:
[1, 0]
[2st, s^2]
If you plug in 0 for s, you get:
[1, 0]
[0, 0]
Which is a rank one matrix. (The rank of a matrix is the the dimension of its row space; i.e. the number of linearly independent row vectors in the matrix) Our vague idea of a 1-dimensional surface coincides exactly with jacobians that have rank 1! Similarly, if a system of equations describes a 2-d surface the matrix has rank 2. For example, at the point s = 0.5 t = 0.5 the jacobian of the above system is:
[1, 0]
[0.5, 0.25]
Which is rank 2, as we'd expect from plotting the surface.
Hurkyl
P.S. bleh just saw where you said you're joking
P.P.S can you do superscripts in 3.0?
