If you are not already familiar with some of the basics of linear algebra, then this section and its subsections are probably not for you. But I wil make a valiant effort at a crash course anyway.
Linear algebra is one of those math subjects that begins with an innocent question, and then develops within two short pages into n-dimensional vector spaces where you cannot see the forest for the trees.
The innocent question is : if three apples and two oranges cost $5, and three apples and three oranges cost $6, how much does an apple cost ? The answer, of course, is $1. Subtracting the first "equation" from the second shows that one orange costs $1, and entering that back into the first equation says that three apples cost $3.
It's algebra because you can temporarily write the numbers as letters until you know their values, and it is linear because everything is one-on-one. There are no squares, cubes or worse involved, just plain multiplication of one or more variables by one or more factors. It is hard to believe that from such humble beginnings, an abstract theory has evolved that is useful from finite element methods to aircraft stability and control, but there it is.
You cannot pinpoint the price of an orange from just the first equation. But if you have a second equation, you can "eliminate" the apples by subtracting the two equations, maybe with a scale factor to make the number of apples equal in both ( this is called "Gauss elimination" ). If you have N unknowns, you need N equations.
Allowing any number of equations ("N") instead of just two, is the first level of abstraction right there.
Taking Gulliver steps, we can first write the apples and oranges as follows :
3 * x + 3 * y = 5
3 * x + 4 * y = 6
Here x and y are the unknown prices of apples and oranges. We can make this more general by replacing the "known" numbers by symbols too :
a * x + b * y = e
c * x + d * y = f
I remember how in high school it baffled me that the a, b, c and d seemed to belong to a different "family" than the x and the y. Why would one letter be "known" and the other "unknown" ?
As it turns out, I was right about that. You can juggle with these numbers any way you like. You can fill in one set and determine the other, in any mixture you like. The only thing that remains true is that from N equations, you can never find more than N "unknown" values. This truth holds out in all of mathematics and beyond.
- parallel lines. Useless duplication. Subtle connections.
Matrices are just a shorthand to note down the simultaneous linear equations. The name means something like a grid to collects numbers in.
The equations above in matrix notation are :
If the rules of matrix and vector multiplication are really new to you, then now would be the time to look up an introductory text on linear algebra. Having grouped the numbers like this, we can give names to each block of numbers. We can write :
A . x = b
The "matrix" A is the square block of numbers. It does not have to be square, but in our applications it usually is, because we try to have the same number of equations as unknowns. The columns of numbers x and b are called "vectors".
There, now I've done it myself. We have moved from a very simple problem to a very abstract concept.
- parallel with directions in space. And rotations and scalings.
- TBW.