A Transformation moves an object. The move could be a rotation, or a translation (moving forward is an example of a translation).
The object isn’t changed by the transformation, the numbers representing the object change. For example, a vector in one coordinate system has the numbers (0,5) and in another coordinate system it has the numbers (3,4).
We will show several examples below of the same vector.
The transformation can be a rotation that takes the vector from the old coordinate system to the new coordinate system.
The transformation can take the vector from one scale of measurement to another. If the first vector shown has units of inches, then the transformation shown below takes it to a system with units of centimeters.
We could do a transformation that takes a vector from Cartesian Coordinates and puts it on Polar Coordinates. We want to do this, it will look “cool”, but it will take some time.
These transformations can be done on the above vector using matrices. Scroll up to the first example. It is (3,4). Think back to the Pythagorean Theorem: this vector has a length of 5.
The first transformation takes this vector (5,0). This transformation can be done with the matrix shown below:
A rotation of 36.87 deg will move the x-axis under the vector.
- sin (36.87) = 0.6000
- cos (36.87) = 0.8000
Appendix A
At some point you’ll run into a discussion in math where it helps to realize that there are two ways of looking at motion, and the math can’t tell you to use one over the other. Consider the two sentences:
- I’m moving on the road with the velocity of 100 mph east.
- The road is moving under me with a velocity of 100 mph west.
The notion of “relative” plays into the above thinking. Take that same thinking to the two sentences below:
- A matrix rotates an object 30 degrees counterclockwise relative to the graph paper.
- A matrix rotates the graph paper 30 degrees clockwise relative to the object.
Rob Sterling 