In my last post, I showed how to determine an unknown matrix for most transformations in the xy-plane and suggested that they held even more information.
Given a pre-image set of points which can be connected to enclose one or more areas with either clockwise or counterclockwise orientation. If a transformation T represented by matrix is applied to the pre-image points, then the determinant of , , tells you two things about the image points.
- The area enclosed by similarly connecting the image points is times the area enclosed by the pre-image points, and
- The orientation of the image points is identical to that of the pre-image if , but is reversed if . If , then the image area is 0 by the first property, and any question about orientation is moot.
In other words, is the area scaling factor from the pre-image to the image (addressing the second half of CCSSM Standard NV-M 12 on page 61 here), and the sign of indicates whether the pre-image and image have the same or opposite orientation, a property beyond the stated scope of the CCSSM.
Example 1: Interpret for the matrix representing a reflection over the x-axis, .
From here, . The magnitude of this is 1, indicating that the area of an image of an object reflected over the line is 1 times the area of the pre-image—an obviously true fact because reflections preserve area.
Also, indicating that the orientation of the reflection image is reversed from that of its pre-image. This, too, must be true because reflections reverse orientation.
Example 2: Interpret for the matrix representing a scale change that doubles x-coordinates and triples y-coordinates, .
For this matrix, , indicating that the image’s area is 6 times that of its pre-image area, while both the image and pre-image have the same orientation. Both of these facts seem reasonable if you imagine a rectangle as a pre-image. Doubling the base and tripling the height create a new rectangle whose area is six times larger. As no flipping is done, orientation should remain the same.
Example 3 & a Pythagorean Surprise: What should be true about for the transformation matrix representing a generic rotation of units around the origin, ?
Rotations preserve area without reversing orientation, so should be +1. Using this fact and computing the determinant gives
In a generic right triangle with hypotenuse C, leg A adjacent to acute angle , and another leg B, this equation is equivalent to , or , the Pythagorean Theorem. There are literally hundreds of proofs of this theorem, and I suspect this proof has been given sometime before, but I think this is a lovely derivation of that mathematical hallmark.
Conclusion: While it seems that these two properties about the determinants of transformation matrices are indeed true for the examples shown, mathematicians hold out for a higher standard. I’ll offer a proof of both properties in my next post.