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 with respect to the Euclidean distance function. 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 $R_{l}.$

    Solution

    This only required `plugging in' to the formula and applying the resulting matrix to the given points. For instance
    MATH
    and
    MATH

  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.

    Proof

    Let MATH and MATH and consider the reflection represented by the matrix
    MATH
    $.$ Now let
    MATH

    Now
    MATH

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

    Proof

    Let $S,$ $T$ be two isometries with respect to a distance $d.$ We claim that $S\circ T$ is an isometry. Let $P$ and $Q$ be two points and note that
    MATH
    where the first line follows from the definition of composition and the succeeding lines follow from the definition of an isometry. As $P$ and $Q$ were arbitrary, $S\circ T$ 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

    Proof

    Let
    MATH
    and
    MATH
    be two matrices representing reflections. Now observe that
    MATH
    where $e=ac+bd$ and $f=bc-ad.$ Now observe that
    MATH
    and hence the product of
    MATH
    and
    MATH
    is a `rotation matrix'.

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