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.
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\(A=\left[\begin{array}{cc}{1}&{2}\\{3}&{4}\end{array}\right]\)
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\(A=\left[\begin{array}{cc}{-1}&{2}\\{1}&{2}\end{array}\right]\)
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\(A=\left[\begin{array}{cc}{1}&{2}\\{1}&{2}\end{array}\right]\)
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\(A=\left[\begin{array}{cc}{2}&{-1}\\{0}&{0}\end{array}\right]\)
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.
- vertical shear by a factor of \(2\)
- horizontal shear by a factor of \(2\)
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\(A=\left[\begin{array}{cc}{5}&{2}\\{2}&{1}\end{array}\right]\)
- horizontal shear by a factor of \(2\)
- vertical shear by a factor of \(2\)
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\(A=\left[\begin{array}{cc}{1}&{2}\\{2}&{5}\end{array}\right]\)
- horizontal stretch by a factor of \(3\)
- reflection across the line \(y = x\)
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\(A=\left[\begin{array}{cc}{0}&{1}\\{3}&{0}\end{array}\right]\)
- counterclockwise rotation by an angle of \(45^{\circ}\)
- vertical stretch by a factor of \(1/2\)
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\(A=\left[\begin{array}{cc}{0.707}&{-0.707}\\{0.354}&{0.354}\end{array}\right]\)
- clockwise rotation by an angle of \(90^{\circ}\)
- horizontal reflection across the \(y\) axis
- vertical shear by a factor of \(1\)
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\(A=\left[\begin{array}{cc}{0}&{-1}\\{-1}&{-1}\end{array}\right]\)
- vertical reflection across the \(x\) axis
- horizontal reflection across the \(y\) axis
- diagonal reflection across the line \(y = x\)
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\(A=\left[\begin{array}{cc}{0}&{-1}\\{-1}&{0}\end{array}\right]\)
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.
- a horizontal reflection across the \(y\) axis, followed by a vertical reflection across the \(x\) axis, compared to
- a counterclockwise rotation of \(180^{\circ}\)
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Yes, these are the same; the transformation matrix in each is \(\left[\begin{array}{cc}{-1}&{0}\\{0}&{-1}\end{array}\right]\).
- 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 \(2\)
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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 \(\left[\begin{array}{cc}{1}&{0}\\{0}&{2}\end{array}\right]\).
- 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
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Yes, these are the same. Each produces the transformation matrix \(\left[\begin{array}{cc}{1/2}&{0}\\{0}&{3}\end{array}\right]\).
- 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 \(y = x\).
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Yes, these are the same. Each produces the transformation matrix \(\left[\begin{array}{cc}{0}&{1}\\{-1}&{0}\end{array}\right]\).