4.03d Linear transformations 2D: reflection, rotation, enlargement, shear

1 Transformation A is represented by matrix \(\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and transformation B is represented by matrix \(\mathbf { B } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 3 \end{array} \right)\).

1. Describe transformations A and B .
2. Find the matrix for the composite transformation A followed by B .

9 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right) , \mathbf { N } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).

1. The matrix products \(\mathbf { Q } ( \mathbf { M N } )\) and \(( \mathbf { Q M } ) \mathbf { N }\) are identical. What property of matrix multiplication does this illustrate? Find QMN. \(\mathbf { M } , \mathbf { N }\) and \(\mathbf { Q }\) represent the transformations \(\mathrm { M } , \mathrm { N }\) and Q respectively.
2. Describe the transformations \(\mathrm { M } , \mathrm { N }\) and Q . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-4_668_908_788_621} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure}
3. The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 9 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively by the composite transformation N followed by M followed by Q . Draw a diagram showing the image of the triangle after this composite transformation, labelling the image of each point clearly.

9 The matrices \(\mathbf { P } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\) represent transformations \(P\) and \(Q\) respectively.

1. Describe fully the transformations P and Q . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e449d411-aaa9-4167-aa9c-c28d31446d52-4_625_849_470_648} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure} Fig. 9 shows triangle T with vertices \(\mathrm { A } ( 2,0 ) , \mathrm { B } ( 1,2 )\) and \(\mathrm { C } ( 3,1 )\).
   Triangle T is transformed first by transformation P , then by transformation Q .
2. Find the single matrix that represents this composite transformation.
3. This composite transformation maps triangle T onto triangle \(\mathrm { T } ^ { \prime }\), with vertices \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). Calculate the coordinates of \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). T' is reflected in the line \(y = - x\) to give a new triangle, T".
4. Find the matrix \(\mathbf { R }\) that represents reflection in the line \(y = - x\).
5. A single transformation maps \(\mathrm { T } ^ { \prime \prime }\) onto the original triangle, T . Find the matrix representing this transformation.

1

1. Write down the matrix for a rotation of \(90 ^ { \circ }\) anticlockwise about the origin.
2. Write down the matrix for a reflection in the line \(y = x\).
3. Find the matrix for the composite transformation of rotation of \(90 ^ { \circ }\) anticlockwise about the origin, followed by a reflection in the line \(y = x\).
4. What single transformation is equivalent to this composite transformation?

1 You are given that the matrix \(\left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)\) represents a transformation \(A\), and that the matrix \(\left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)\) represents a transformation B .

1. Describe the transformations A and B .
2. Find the matrix representing the composite transformation consisting of A followed by B .
3. What single transformation is represented by this matrix?

9

1. Describe fully the transformation Q , represented by the matrix \(\mathbf { Q }\), where \(\mathbf { Q } = \left( \begin{array} { r l } 0 & 1 \\ - 1 & 0 \end{array} \right)\). The transformation M is represented by the matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { r r } 0 & - 1 \\ 0 & 1 \end{array} \right)\).
2. M maps all points on the line \(y = 2\) onto a single point, P. Find the coordinates of P.
3. M maps all points on the plane onto a single line, \(l\). Find the equation of \(l\).
4. M maps all points on the line \(n\) onto the point ( - 6 , 6). Find the equation of \(n\).
5. Show that \(\mathbf { M }\) is singular. Relate this to the transformation it represents.
6. R is the composite transformation M followed by Q . R maps all points on the plane onto the line \(q\). Find the equation of \(q\).

2 Fig. 2 shows the unit square, OABC , and its image, \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\), after undergoing a transformation. \begin{figure}[h] \end{figure}

1. Write down the matrix \(\mathbf { T }\) representing this transformation. The quadrilateral \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) is reflected in the \(x\)-axis to give a new quadrilateral, \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\).
2. Write down the matrix representing reflection in the \(x\)-axis.
3. Find the single matrix that will transform OABC onto \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\).

9 The triangle ABC has vertices at \(\mathrm { A } ( 0,0 ) , \mathrm { B } ( 0,2 )\) and \(\mathrm { C } ( 4,1 )\). The matrix \(\left( \begin{array} { r r } 1 & - 2 \\ 3 & 0 \end{array} \right)\) represents a transformation T .

1. The transformation \(T\) maps triangle \(A B C\) onto triangle \(A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Find the coordinates of \(A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\). Triangle \(A ^ { \prime } B ^ { \prime } C ^ { \prime }\) is now mapped onto triangle \(A ^ { \prime \prime } B ^ { \prime \prime } C ^ { \prime \prime }\) using the matrix \(\mathbf { M } = \left( \begin{array} { l l } 4 & 0 \\ 0 & 2 \end{array} \right)\).
2. Describe fully the transformation represented by \(\mathbf { M }\).
3. Triangle \(\mathrm { A } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\) is now mapped back onto ABC by a single transformation. Find the matrix representing this transformation.
4. Calculate the area of \(A ^ { \prime \prime } B ^ { \prime \prime } C ^ { \prime \prime }\).

4

1. The curve with Cartesian equation \(\frac { x ^ { 2 } } { c } + \frac { y ^ { 2 } } { d } = 1\) is mapped onto the curve with polar equation \(r = \frac { 10 } { 3 - 2 \cos \theta }\) by a single geometrical transformation. By writing the polar equation as a Cartesian equation in a suitable form, find the values of the constants \(c\) and \(d\).
2. Hence describe the geometrical transformation referred to in part (a).
   [0pt] [1 mark]

8 The \(2 \times 2\) matrix A represents a transformation T which has the following properties.

• The image of the point \(( 0,1 )\) is the point \(( 3,4 )\).
• An object shape whose area is 7 is transformed to an image shape whose area is 35 .
• T has a line of invariant points.
  1. Find a possible matrix for \(\mathbf { A }\).

The transformation S is represented by the matrix \(\mathbf { B }\) where \(\mathbf { B } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)\).

Find the equation of the line of invariant points of S .

Show that any line of the form \(y = x + c\) is an invariant line of S .

6 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \frac { 1 } { 13 } \left( \begin{array} { r r } 5 & 12 \\ 12 & - 5 \end{array} \right)\). You are given that \(\mathbf { A }\) represents the transformation T which is a reflection in a certain straight line. You are also given that this straight line, the mirror line, passes through the origin, \(O\).

1. Explain why there must be a line of invariant points for T . State the geometric significance of this line.
2. By considering the line of invariant points for T , determine the equation of the mirror line. Give your answer in the form \(y = m x + c\). The coordinates of the point \(P\) are \(( 1,5 )\).
3. By considering the image of \(P\) under the transformation T , or otherwise, determine the coordinates of the point on the mirror line which is closest to \(P\).
4. The line with equation \(y = a x + 2\) is an invariant line for T. Determine the value of \(a\).

5 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - \frac { 3 } { 5 } & \frac { 4 } { 5 } \\ \frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)\).

1. The diagram in the Printed Answer Booklet shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { M }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and clearly label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
2. Find the equation of the line of invariant points of this transformation.
3. (a) Find the determinant of \(\mathbf { M }\).
   (b) Describe briefly how this value relates to the transformation represented by \(\mathbf { M }\).

4 The transformations \(T _ { A }\) and \(T _ { B }\) are represented by the matrices \(\mathbf { A }\) and \(\mathbf { B }\) respectively, where \(\mathbf { A } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\)

1. Describe geometrically the single transformation consisting of \(T _ { A }\) followed by \(T _ { B }\).
2. By considering the transformation \(\mathrm { T } _ { \mathrm { A } }\), determine the matrix \(\mathrm { A } ^ { 423 }\). The transformation \(\mathrm { T } _ { \mathrm { C } }\) is represented by the matrix \(\mathbf { C }\), where \(\mathbf { C } = \left( \begin{array} { l l } \frac { 1 } { 2 } & 0 \\ 0 & \frac { 1 } { 3 } \end{array} \right)\). The region \(R\) is defined by the set of points \(( x , y )\) satisfying the inequality \(x ^ { 2 } + y ^ { 2 } \leqslant 36\). The region \(R ^ { \prime }\) is defined as the image of \(R\) under \(\mathrm { T } _ { \mathrm { C } }\).
3. 1. Find the exact area of the region \(R ^ { \prime }\).
   2. Sketch the region \(R ^ { \prime }\), specifying all the points where the boundary of \(R ^ { \prime }\) intersects the coordinate axes.

7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 0.6 & 2.4 \\ - 0.8 & 1.8 \end{array} \right)\).

1. Find \(\operatorname { det } \mathbf { A }\). The matrix A represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.
2. By considering the determinants of these transformations, determine the scale factor of the stretch.
3. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer.
4. Find the angle of rotation.

6 In this question you must show detailed reasoning.
The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).

1. Define the transformation represented by \(\mathbf { A }\).
2. Show that the area of any object shape is invariant under the transformation represented by \(\mathbf { A }\). The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r l } 7 & 2 \\ 21 & 7 \end{array} \right)\). You are given that \(\mathbf { B }\) represents the transformation which is the result of applying the following three transformations in the given order.

7

1. The transformation T is defined by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right]$$
   1. Describe the transformation T geometrically.
   2. Calculate the matrix product \(\mathbf { A } ^ { 2 }\).
   3. Explain briefly why the transformation T followed by T is the identity transformation.
2. The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$$
   1. Calculate \(\mathbf { B } ^ { 2 } - \mathbf { A } ^ { 2 }\).
   2. Calculate \(( \mathbf { B } + \mathbf { A } ) ( \mathbf { B } - \mathbf { A } )\).

2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right] , \mathbf { B } = \left[ \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \\ \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right]$$

1. Calculate:
   1. \(\mathbf { A } + \mathbf { B }\);
   2. \(\mathbf { B A }\).
2. Describe fully the geometrical transformation represented by each of the following matrices:
   1. \(\mathbf { A }\);
   2. \(\mathbf { B }\);
   3. \(\mathbf { B A }\).

5 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } k & k \\ k & - k \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } - k & k \\ k & k \end{array} \right]$$ where \(k\) is a constant.

1. Find, in terms of \(k\) :
   1. \(\mathbf { A } + \mathbf { B }\);
   2. \(\mathbf { A } ^ { 2 }\).
2. Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = \mathbf { A } ^ { 2 } + \mathbf { B } ^ { 2 }\).
3. It is now given that \(k = 1\).
   1. Describe the geometrical transformation represented by the matrix \(\mathbf { A } ^ { 2 }\).
   2. The matrix \(\mathbf { A }\) represents a combination of an enlargement and a reflection. Find the scale factor of the enlargement and the equation of the mirror line of the reflection.

3

1. Write down the \(2 \times 2\) matrix corresponding to each of the following transformations:
   1. a rotation about the origin through \(90 ^ { \circ }\) clockwise;
   2. a rotation about the origin through \(180 ^ { \circ }\).
2. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { r r } 2 & 4 \\ - 1 & - 3 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { l l } - 2 & 1 \\ - 4 & 3 \end{array} \right]$$
   1. Calculate the matrix \(\mathbf { A B }\).
   2. Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = k \mathbf { I }\), where \(\mathbf { I }\) is the identity matrix, for some integer \(k\).
3. Describe the single geometrical transformation, or combination of two geometrical transformations, represented by each of the following matrices:
   1. \(\mathbf { A } + \mathbf { B }\);
   2. \(( \mathbf { A } + \mathbf { B } ) ^ { 2 }\);
   3. \(( \mathbf { A } + \mathbf { B } ) ^ { 4 }\).

8 The diagram below shows a rectangle \(R _ { 1 }\) which has vertices \(( 0,0 ) , ( 3,0 ) , ( 3,2 )\) and \(( 0,2 )\).

1. On the diagram, draw:
   1. the image \(R _ { 2 }\) of \(R _ { 1 }\) under a rotation through \(90 ^ { \circ }\) clockwise about the origin;
   2. the image \(R _ { 3 }\) of \(R _ { 2 }\) under the transformation which has matrix $$\left[ \begin{array} { l l } 4 & 0 \\ 0 & 2 \end{array} \right]$$
2. Find the matrix of:
   1. the rotation which maps \(R _ { 1 }\) onto \(R _ { 2 }\);
   2. the combined transformation which maps \(R _ { 1 }\) onto \(R _ { 3 }\). \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-5_913_910_1228_598}

6

1. The matrix \(\mathbf { X }\) is defined by \(\left[ \begin{array} { l l } 1 & 2 \\ 3 & 0 \end{array} \right]\).
   1. Given that \(\mathbf { X } ^ { 2 } = \left[ \begin{array} { c c } m & 2 \\ 3 & 6 \end{array} \right]\), find the value of \(m\).
   2. Show that \(\mathbf { X } ^ { 3 } - 7 \mathbf { X } = n \mathbf { I }\), where \(n\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
2. It is given that \(\mathbf { A } = \left[ \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right]\).
   1. Describe the geometrical transformation represented by \(\mathbf { A }\).
   2. The matrix \(\mathbf { B }\) represents an anticlockwise rotation through \(45 ^ { \circ }\) about the origin. Show that \(\mathbf { B } = k \left[ \begin{array} { r r } 1 & - 1 \\ 1 & 1 \end{array} \right]\), where \(k\) is a surd.
   3. Find the image of the point \(P ( - 1,2 )\) under an anticlockwise rotation through \(45 ^ { \circ }\) about the origin, followed by the transformation represented by \(\mathbf { A }\). \(7 \quad\) The variables \(y\) and \(x\) are related by an equation of the form $$y = a x ^ { n }$$ where \(a\) and \(n\) are constants. Let \(Y = \log _ { 10 } y\) and \(X = \log _ { 10 } x\).

1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left[ \begin{array} { l l } 2 & 1 \\ 3 & 8 \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right]$$ The matrix \(\mathbf { M } = \mathbf { A } - 2 \mathbf { B }\).

1. Show that \(\mathbf { M } = n \left[ \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right]\), where \(n\) is a positive integer.
   (2 marks)
2. The matrix \(\mathbf { M }\) represents a combination of an enlargement of scale factor \(p\) and a reflection in a line \(L\). State the value of \(p\) and write down the equation of \(L\).
3. Show that $$\mathbf { M } ^ { 2 } = q \mathbf { I }$$ where \(q\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.

8 [Figure 3, printed on the insert, is provided for use in this question.]
The diagram shows two triangles, \(T _ { 1 }\) and \(T _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{504b79bf-1bcc-4fa7-a7a0-689c21a8b03a-04_866_883_1318_550}

1. Find the matrix of the stretch which maps \(T _ { 1 }\) to \(T _ { 2 }\).
2. The triangle \(T _ { 2 }\) is reflected in the line \(y = x\) to give a third triangle, \(T _ { 3 }\). On Figure 3, draw the triangle \(T _ { 3 }\).
3. Find the matrix of the transformation which maps \(T _ { 1 }\) to \(T _ { 3 }\).

7

1. Using surd forms where appropriate, find the matrix which represents:
   1. a rotation about the origin through \(30 ^ { \circ }\) anticlockwise;
   2. a reflection in the line \(y = \frac { 1 } { \sqrt { 3 } } x\).
2. The matrix \(\mathbf { A }\), where $$\mathbf { A } = \left[ \begin{array} { c c } 1 & \sqrt { 3 } \\ \sqrt { 3 } & - 1 \end{array} \right]$$ represents a combination of an enlargement and a reflection. Find the scale factor of the enlargement and the equation of the mirror line of the reflection.
3. The transformation represented by \(\mathbf { A }\) is followed by the transformation represented by \(\mathbf { B }\), where $$\mathbf { B } = \left[ \begin{array} { c c } \sqrt { 3 } & - 1 \\ 1 & \sqrt { 3 } \end{array} \right]$$ Find the matrix of the combined transformation and give a full geometrical description of this combined transformation.