5. Matrices
Hovedinnhold
Let us look at the two equations
y1 = 2x0 + 3y0
x1 = 3x0 - y0
If we plug in a coordinate pair (x0, y0), then we get a new coordinate pair (x1, y1). We can represent this transformation in a simpler form by just writing the coefficients in a matrix.
A =
| 2 | 3 |
| 3 | -1 |
If we continue now and make the transformation
y2 = x1 + 2y1
x2 = -2x1 - y1
then we get a new pair (x2, y2). The transformation can again be represented by the matrix
B =
| 1 | 2 |
| -2 | -1 |
If we drop the middle step, the calculation of (x1, y1), and go directly from (x0, y0) to (x2, y2), then one can figure out that the transformation is
y2 = 8x0 + y0
x2 = -7x0 - 5y0
The matrix here is
C =
| 8 | 1 |
| -7 | -5 |
and we call this the product of the matrices B and A. We write
C = B × A.
We can also add two matrices together by adding the numbers position by position
A + B =
| 3 | 5 |
| 1 | -2 |
This gives us an algebraic system where we can add, subtract and multiply. But we can not always divide. Such sets resemble the integers. But there is a fundamental difference.
Read on about commutative versus noncommutative systems.