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5.1.1: Exercises 5.1

  • Page ID
    70788
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    In Exercises \(\PageIndex{1}\) - \(\PageIndex{4}\), a sketch of transformed unit square is given. Find the matrix \(A\) that performs this transformation.

    Exercise \(\PageIndex{1}\)

    clipboard_e2895a3158a3263c1c6bbb7b29f87703b.png

    Answer

    \(A=\left[\begin{array}{cc}{1}&{2}\\{3}&{4}\end{array}\right]\)

    Exercise \(\PageIndex{2}\)

    clipboard_e9a13b70e3e27d7531429f3dae7fdf330.png

    Answer

    \(A=\left[\begin{array}{cc}{-1}&{2}\\{1}&{2}\end{array}\right]\)

    Exercise \(\PageIndex{3}\)

    clipboard_ee821ea81eef56e37593704bd0d24854a.png

    Answer

    \(A=\left[\begin{array}{cc}{1}&{2}\\{1}&{2}\end{array}\right]\)

    Exercise \(\PageIndex{4}\)

    clipboard_e434bf5cdd4a31c6b5671db2de6967e2d.png

    Answer

    \(A=\left[\begin{array}{cc}{2}&{-1}\\{0}&{0}\end{array}\right]\)

    In Exercises \(\PageIndex{5}\) – \(\PageIndex{10}\), a list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane.

    Exercise \(\PageIndex{5}\)
    1. vertical shear by a factor of \(2\)
    2. horizontal shear by a factor of \(2\)
    Answer

    \(A=\left[\begin{array}{cc}{5}&{2}\\{2}&{1}\end{array}\right]\)

    Exercise \(\PageIndex{6}\)
    1. horizontal shear by a factor of \(2\)
    2. vertical shear by a factor of \(2\)
    Answer

    \(A=\left[\begin{array}{cc}{1}&{2}\\{2}&{5}\end{array}\right]\)

    Exercise \(\PageIndex{7}\)
    1. horizontal stretch by a factor of \(3\)
    2. reflection across the line \(y = x\)
    Answer

    \(A=\left[\begin{array}{cc}{0}&{1}\\{3}&{0}\end{array}\right]\)

    Exercise \(\PageIndex{8}\)
    1. counterclockwise rotation by an angle of \(45^{\circ}\)
    2. vertical stretch by a factor of \(1/2\)
    Answer

    \(A=\left[\begin{array}{cc}{0.707}&{-0.707}\\{0.354}&{0.354}\end{array}\right]\)

    Exercise \(\PageIndex{9}\)
    1. clockwise rotation by an angle of \(90^{\circ}\)
    2. horizontal reflection across the \(y\) axis
    3. vertical shear by a factor of \(1\)
    Answer

    \(A=\left[\begin{array}{cc}{0}&{-1}\\{-1}&{-1}\end{array}\right]\)

    Exercise \(\PageIndex{10}\)
    1. vertical reflection across the \(x\) axis
    2. horizontal reflection across the \(y\) axis
    3. diagonal reflection across the line \(y = x\)
    Answer

    \(A=\left[\begin{array}{cc}{0}&{-1}\\{-1}&{0}\end{array}\right]\)

    In Exercises \(\PageIndex{11}\) – \(\PageIndex{14}\), two sets of transformations are given. Sketch the transformed unit square under each set of transformations. Are the transformations the same? Explain why/why not.

    Exercise \(\PageIndex{11}\)
    1. a horizontal reflection across the \(y\) axis, followed by a vertical reflection across the \(x\) axis, compared to
    2. a counterclockwise rotation of \(180^{\circ}\)
    Answer

    Yes, these are the same; the transformation matrix in each is \(\left[\begin{array}{cc}{-1}&{0}\\{0}&{-1}\end{array}\right]\).

    Exercise \(\PageIndex{12}\)
    1. a horizontal stretch by a factor of \(2\) followed by a reflection across the line \(y = x\), compared to
    2. a vertical stretch by a factor of \(2\)
    Answer

    No, these are different. The first produces a transformation matrix \(\left[\begin{array}{cc}{0}&{1}\\{2}&{0}\end{array}\right]\), which the second produces \(\left[\begin{array}{cc}{1}&{0}\\{0}&{2}\end{array}\right]\).

    Exercise \(\PageIndex{13}\)
    1. a horizontal stretch by a factor of \(1/2\) followed by a vertical stretch by a factor of \(3\), compared to
    2. the same operations but in opposite order
    Answer

    Yes, these are the same. Each produces the transformation matrix \(\left[\begin{array}{cc}{1/2}&{0}\\{0}&{3}\end{array}\right]\).

    Exercise \(\PageIndex{14}\)
    1. a reflection across the line \(y = x\) followed by a reflection across the \(x\) axis, compared to
    2. a reflection across the the \(y\) axis, followed by a reflection across the line \(y = x\).
    Answer

    Yes, these are the same. Each produces the transformation matrix \(\left[\begin{array}{cc}{0}&{1}\\{-1}&{0}\end{array}\right]\).


    This page titled 5.1.1: Exercises 5.1 is shared under a CC BY-NC 3.0 license and was authored, remixed, and/or curated by Gregory Hartman et al..

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