5.1.1: Exercises 5.1

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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 $\PageIndex{5}$

vertical shear by a factor of $2$
horizontal shear by a factor of $2$

Answer: $A=\left[\begin{array}{cc}{5}&{2}\\{2}&{1}\end{array}\right]$

Exercise $\PageIndex{6}$

horizontal shear by a factor of $2$
vertical shear by a factor of $2$

Answer: $A=\left[\begin{array}{cc}{1}&{2}\\{2}&{5}\end{array}\right]$

Exercise $\PageIndex{7}$

horizontal stretch by a factor of $3$
reflection across the line $y = x$

Answer: $A=\left[\begin{array}{cc}{0}&{1}\\{3}&{0}\end{array}\right]$

Exercise $\PageIndex{8}$

counterclockwise rotation by an angle of $45^{\circ}$
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 $\PageIndex{9}$

clockwise rotation by an angle of $90^{\circ}$
horizontal reflection across the $y$ axis
vertical shear by a factor of $1$

Answer: $A=\left[\begin{array}{cc}{0}&{-1}\\{-1}&{-1}\end{array}\right]$

Exercise $\PageIndex{10}$

vertical reflection across the $x$ axis
horizontal reflection across the $y$ axis
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 $\PageIndex{11}$

a horizontal reflection across the $y$ axis, followed by a vertical reflection across the $x$ axis, compared to
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 $\PageIndex{12}$

a horizontal stretch by a factor of $2$ followed by a reflection across the line $y = x$, compared to
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 $\PageIndex{13}$

a horizontal stretch by a factor of $1/2$ followed by a vertical stretch by a factor of $3$, compared to
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 $\PageIndex{14}$

a reflection across the line $y = x$ followed by a reflection across the $x$ axis, compared to
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 $\left[\begin{array}{cc}{0}&{1}\\{-1}&{0}\end{array}\right]$.

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