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

8 The matrix \(\mathbf { T }\) is given by \(\mathbf { T } = \left( \begin{array} { r r } 2 & 0 \\ 0 & - 2 \end{array} \right)\).

1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { T }\). [3]
2. The transformation represented by matrix \(\mathbf { T }\) is equivalent to a transformation \(A\), followed by a transformation B. Give geometrical descriptions of possible transformations A and B, and state the matrices that represent them.

9 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r } 0 & 3 \\ - 1 & 0 \end{array} \right)\).

1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\). The transformation represented by \(\mathbf { C }\) is equivalent to a rotation, R , followed by another transformation, S.
2. Describe fully the rotation R and write down the matrix that represents R .
3. Describe fully the transformation S and write down the matrix that represents S .

1 The transformation S is a shear with the \(y\)-axis invariant (i.e. a shear parallel to the \(y\)-axis). It is given that the image of the point \(( 1,1 )\) is the point \(( 1,0 )\).

1. Draw a diagram showing the image of the unit square under the transformation S .
2. Write down the matrix that represents S .

2 The transformation S is a shear parallel to the \(x\)-axis in which the image of the point ( 1,1 ) is the point \(( 0,1 )\).

1. Draw a diagram showing the image of the unit square under S .
2. Write down the matrix that represents S .

9

1. Write down the matrix, \(\mathbf { A }\), that represents an enlargement, centre ( 0,0 ), with scale factor \(\sqrt { 2 }\).
2. The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \\ - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { B }\).
3. Given that \(\mathbf { C } = \mathbf { A B }\), show that \(\mathbf { C } = \left( \begin{array} { r r } 1 & 1 \\ - 1 & 1 \end{array} \right)\).
4. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { C }\).
5. Write down the determinant of \(\mathbf { C }\) and explain briefly how this value relates to the transformation represented by \(\mathbf { C }\).

7 Describe fully the geometrical transformation represented by each of the following matrices:

1. \(\left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)\),
2. \(\left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\),
3. \(\left( \begin{array} { l l } 1 & 0 \\ 0 & 6 \end{array} \right)\),
4. \(\left( \begin{array} { r r } 0.8 & 0.6 \\ - 0.6 & 0.8 \end{array} \right)\).

7

1. Find the matrix that represents a rotation through \(90 ^ { \circ }\) clockwise about the origin.
2. Find the matrix that represents a reflection in the \(x\)-axis.
3. Hence find the matrix that represents a rotation through \(90 ^ { \circ }\) clockwise about the origin, followed by a reflection in the \(x\)-axis.
4. Describe a single transformation that is represented by your answer to part (iii).

7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 1 & - 2 \\ 2 & 1 \end{array} \right)\).

1. Draw a diagram showing the unit square and its image under the transformation represented by \(\mathbf { A }\).
2. The value of \(\operatorname { det } \mathbf { A }\) is 5 . Show clearly how this value relates to your diagram in part (i). A represents a sequence of two elementary geometrical transformations, one of which is a rotation \(R\).
3. Determine the angle of \(R\), and describe the other transformation fully.
4. State the matrix that represents \(R\), giving the elements in an exact form.

9 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6 \\ 0.6 & - 0.8 \end{array} \right)\).

1. Calculate \(\mathbf { M } ^ { 2 }\). You are now given that the matrix \(M\) represents a reflection in a line through the origin.
2. Explain how your answer to part (i) relates to this information.
3. By investigating the invariant points of the reflection, find the equation of the mirror line.
4. Describe fully the transformation represented by the matrix \(\mathbf { P } = \left( \begin{array} { c c } 0.8 & - 0.6 \\ 0.6 & 0.8 \end{array} \right)\).
5. A composite transformation is formed by the transformation represented by \(\mathbf { P }\) followed by the transformation represented by \(\mathbf { M }\). Find the single matrix that represents this composite transformation.
6. The composite transformation described in part ( \(\mathbf { v }\) ) is equivalent to a single reflection. What is the equation of the mirror line of this reflection?

9 A transformation T acts on all points in the plane. The image of a general point P is denoted by \(\mathrm { P } ^ { \prime }\). \(\mathrm { P } ^ { \prime }\) always lies on the line \(y = 2 x\) and has the same \(y\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h] \end{figure}

1. Write down the image of the point \(( 10,50 )\) under transformation T .
2. P has coordinates \(( x , y )\). State the coordinates of \(\mathrm { P } ^ { \prime }\).
3. All points on a particular line \(l\) are mapped onto the point \(( 3,6 )\). Write down the equation of the line \(l\).
4. In part (iii), the whole of the line \(l\) was mapped by T onto a single point. There are an infinite number of lines which have this property under T. Describe these lines.
5. For a different set of lines, the transformation T has the same effect as translation parallel to the \(x\)-axis. Describe this set of lines.
6. Find the \(2 \times 2\) matrix which represents the transformation.
7. Show that this matrix is singular. Relate this result to the transformation.

3 The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 3 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively under the transformation represented by the matrix \(\mathbf { M } = \left( \begin{array} { l l } 2 & 0 \\ 0 & \frac { 1 } { 2 } \end{array} \right)\). \begin{figure}[h] \end{figure}

1. Draw a diagram showing the image of the triangle after the transformation, labelling the image of each point clearly.
2. Describe fully the transformation represented by the matrix \(\mathbf { M }\).

9 A transformation T acts on all points in the plane. The image of a general point P is denoted by \(\mathrm { P } ^ { \prime }\). \(\mathrm { P } ^ { \prime }\) always lies on the line \(y = x\) and has the same \(x\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h] \end{figure}

1. Write down the image of the point ( \(- 3,7\) ) under transformation T .
2. Write down the image of the point \(( x , y )\) under transformation T .
3. Find the \(2 \times 2\) matrix which represents the transformation.
4. Describe the transformation M represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).
5. Find the matrix representing the composite transformation of T followed by M .
6. Find the image of the point \(( x , y )\) under this composite transformation. State the equation of the line on which all of these images lie.

1

1. Write down the matrix for reflection in the \(y\)-axis.
2. Write down the matrix for enlargement, scale factor 3, centred on the origin.
3. Find the matrix for reflection in the \(y\)-axis, followed by enlargement, scale factor 3 , centred on the origin.

1. The triangle \(T\) has vertices \(A ( 2,1 ) , B ( 2,3 )\) and \(C ( 0,1 )\).

The triangle \(T ^ { \prime }\) is the image of \(T\) under the transformation represented by the matrix $$\mathbf { P } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)$$

1. Find the coordinates of the vertices of \(T ^ { \prime }\)
2. Describe fully the transformation represented by \(\mathbf { P }\) The \(2 \times 2\) matrix \(\mathbf { Q }\) represents a reflection in the \(x\)-axis and the \(2 \times 2\) matrix \(\mathbf { R }\) represents a rotation through \(90 ^ { \circ }\) anticlockwise about the origin.
3. Write down the matrix \(\mathbf { Q }\) and the matrix \(\mathbf { R }\)
4. Find the matrix \(\mathbf { R Q }\)
5. Give a full geometrical description of the single transformation represented by the answer to part (d).

6

1. The transformation P is represented by the matrix \(\left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\). Give a geometrical description of transformation P .
2. The transformation Q is represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\). Give a geometrical description of transformation Q.
3. The transformation R is equivalent to transformation P followed by transformation Q . Find the matrix that represents R .
4. Give a geometrical description of the single transformation that is represented by your answer to part (iii).

5

1. The transformation T is represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\). Give a geometrical description of T .
2. The transformation T is equivalent to a reflection in the line \(y = - x\) followed by another transformation S . Give a geometrical description of S and find the matrix that represents S .

10 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).

1. Find \(\mathbf { M } ^ { 2 }\) and \(\mathbf { M } ^ { 3 }\).
2. Hence suggest a suitable form for the matrix \(\mathbf { M } ^ { n }\).
3. Use induction to prove that your answer to part (ii) is correct.
4. Describe fully the single geometrical transformation represented by \(\mathbf { M } ^ { 10 }\).

5

1. Find the matrix that represents a reflection in the line \(y = - x\).
2. The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 4 \end{array} \right)\).
   1. Describe fully the geometrical transformation represented by \(\mathbf { C }\).
   2. State the value of the determinant of \(\mathbf { C }\) and describe briefly how this value relates to the transformation represented by \(\mathbf { C }\).

8 The matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { l l } 3 & 2 \\ 1 & 1 \end{array} \right)\).

1. Draw a diagram showing the image of the unit square under the transformation represented by \(\mathbf { C }\). The transformation represented by \(\mathbf { C }\) is equivalent to a transformation S followed by another transformation T.
2. Given that S is a shear with the \(y\)-axis invariant in which the image of the point ( 1,1 ) is ( 1,2 ), write down the matrix that represents \(S\).
3. Find the matrix that represents transformation T and describe fully the transformation T .

8 The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { l l } 0 & 3 \\ 3 & 0 \end{array} \right)\).

1. The diagram in the printed answer book shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { X }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
2. The transformation represented by \(\mathbf { X }\) is equivalent to a transformation A , followed by a transformation B. Give geometrical descriptions of possible transformations A and B and state the matrices that represent them.

9

1. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { X }\).
2. The matrix \(\mathbf { Z }\) is given by \(\mathbf { Z } = \left( \begin{array} { c c } \frac { 1 } { 2 } & \frac { 1 } { 2 } ( 2 + \sqrt { 3 } ) \\ - \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } ( 1 - 2 \sqrt { 3 } ) \end{array} \right)\). The transformation represented by \(\mathbf { Z }\) is equivalent to the transformation represented by \(\mathbf { X }\), followed by another transformation represented by the matrix \(\mathbf { Y }\). Find \(\mathbf { Y }\).
3. Describe fully the geometrical transformation represented by \(\mathbf { Y }\).

4

1. Find the matrix that represents a shear with the \(y\)-axis invariant, the image of the point \(( 1,0 )\) being the point \(( 1,4 )\).
2. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \\ - \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right)\).
   1. Describe fully the geometrical transformation represented by \(\mathbf { X }\).
   2. Find the value of the determinant of \(\mathbf { X }\) and describe briefly how this value relates to the transformation represented by \(\mathbf { X }\).

6 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } 0 & 2 \\ - 1 & 0 \end{array} \right)\).

1. The diagram in the Printed Answer Book 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 label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\), indicating clearly the coordinates of \(A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\).
2. The transformation represented by \(\mathbf { M }\) is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a possible pair of transformations P and Q and state the matrices that represent them.

7 The matrix \(\left( \begin{array} { l l } 1 & 3 \\ 0 & 1 \end{array} \right)\) represents a transformation P .

1. Describe fully the transformation P . The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - 3 & - 1 \\ - 1 & 0 \end{array} \right)\).
2. Given that \(\mathbf { M }\) represents transformation Q followed by transformation P , find the matrix that represents the transformation Q and describe fully the transformation Q .

9 The matrix \(\mathbf { R }\) is \(\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).

1. Explain in terms of transformations why \(\mathbf { R } ^ { 4 } = \mathbf { I }\).
2. Describe the transformation represented by \(\mathbf { R } ^ { - 1 }\) and write down the matrix \(\mathbf { R } ^ { - 1 }\).
3. \(\mathbf { S }\) is the matrix representing rotation through \(60 ^ { \circ }\) anticlockwise about the origin. Find \(\mathbf { S }\).
4. Write down the smallest positive integers \(m\) and \(n\) such that \(\mathbf { S } ^ { m } = \mathbf { R } ^ { n }\), explaining your answer in terms of transformations.
5. Find \(\mathbf { R S }\) and explain in terms of transformations why \(\mathbf { R S } = \mathbf { S R }\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}