Transformations II and a Pythagorean Surprise

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 [T]= \left[ \begin{array}{cc} A & C \\ B & D \end{array}\right] is applied to the pre-image points, then the determinant of [T], det[T]=AD-BC, tells you two things about the image points.

  1. The area enclosed by similarly connecting the image points is \left| det[T] \right| times the area enclosed by the pre-image points, and
  2. The orientation of the image points is identical to that of the pre-image if det[T]>0, but is reversed if det[T]<0.  If det[T]=0, then the image area is 0 by the first property, and any question about orientation is moot.

In other words, det[T] 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 det[T] 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 det[T] for the matrix representing a reflection over the x-axis, [T]=\left[ r_{x-axis} \right] =\left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right].

From here, det[T]=-1.  The magnitude of this is 1, indicating that the area of an image of an object reflected over the line y=x is 1 times the area of the pre-image—an obviously true fact because reflections preserve area.

Also, det \left[ r_{x-axis} \right]<0 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 det[T] for the matrix representing a scale change that doubles x-coordinates and triples y-coordinates, [T]=\left[ S_{2,3} \right] =\left[ \begin{array}{cc} 2 & 0 \\ 0 & 3 \end{array} \right].

For this matrix, det[T]=+6, 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  det[T] for the transformation matrix representing a generic rotation of \theta units around the origin,  [T]=\left[ R_\theta \right] = \left[ \begin{array}{cc} cos( \theta ) & -sin( \theta ) \\ sin( \theta ) & cos( \theta ) \end{array} \right] ?

Rotations preserve area without reversing orientation, so det\left[ R_\theta \right] should be +1.  Using this fact and computing the determinant gives

det \left[ R_\theta \right] = cos^2(\theta ) + sin^2(\theta )=+1 .

In a generic right triangle with hypotenuse C, leg A adjacent to acute angle \theta , and another leg B, this equation is equivalent to \left( \frac{A}{C} \right) ^2 + \left( \frac{B}{C} \right) ^2 = 1 , or A^2+B^2=C^2, 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.

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2 responses to “Transformations II and a Pythagorean Surprise

  1. Wow. Very cool stuff. I never really thought about the meaning of a determinant from a transformation point of view.

  2. Pingback: Transformations III | CAS Musings

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