Progress Report: Matrices and Transformations


In this progress report, we will focus on the connection between transformations and matrices. Any linear mapping from $\QTR{Bbb}{R}^{2}$ into itself can be represented as $2\times 2$ matrix. It is a nontrivial result that a mapping that preserves distance and maps the origin to itself is a linear map. (In the general case, this result is due to Mazur and Ulam.) The general topic we would like to look at is the connections between matrices and distance preserving maps. A distance preserving map is often called an isometry or rigid motion.

In class we established that

  1. A rotation of $\theta $ degrees around the origin can be associated with the matrix
    MATH
    Let $M_{\theta }$ denote a rotation of $\theta $ degrees around the origin; what this result says is that if MATH is a point in $\QTR{Bbb}{R}^{2}$ and MATH then MATH Keeping the identity MATH in mind, we can also think of these matrices as being in the form
    MATH
    where $a^{2}+b^{2}=1.$

  2. If a line through the origin makes an angle of $\theta $ degrees with the $x$-axis, the reflection $R_{l}$ can be associated with the matrix
    MATH
    Let $R_{l}$ denote the reflection with axis $l;$ what this result says is that if MATH is a point in $\QTR{Bbb}{R}^{2}$ and MATH then MATH Keeping the identity MATH in mind, we can also think of these matrices being in the form
    MATH
    where $a^{2}+b^{2}=1.$

One thing we didn't do was establish that rotations and reflections are isometries. You will do that later in this progress report.


Questions:

  1. Using the results above, find the matrices that would be associated with a rotation $M_{30}$ of $30$ degrees and a reflection about $l,$where $l$ makes an angle of $30$ degrees with the $x$-axis. Use the matrices to find the image of MATH MATH and MATH under $M_{30}$ and $R_{l}.$

  2. Use the reflection and rotation feature of GSP to confirm your answers to question 1.

  3. Using the definition of the Euclidean distance function, verify that reflections are distance preserving. (This is just a bit of algebra, but it is worth doing.)

  4. Give a short proof, with no algebra, that the composition of two isometries is an isometry.

  5. Using the results above, show that if $l$ and $m$ are two lines through the origin, then $R_{m}\circ R_{l}$ is a rotation. Do this by showing that the product of two matrices of the form
    MATH
    is a matrix of the form
    MATH
    $.$

  6. Group Process: You are (finally) in a new group! What practices did you find beneficial in your old group and will try to bring to this group?

This document created by Scientific WorkPlace 4.0.