Linear transformations

1 The transformation R of the plane is reflection in the line \(x = 0\).

1. Write down the matrix \(\mathbf { M }\) associated with R .
2. Find \(\mathbf { M } ^ { 2 }\).
3. Interpret the result of part (b) in terms of the transformation \(R\).

1 A reflection is represented by the matrix \(\left[ \begin{array} { c c } 1 & 0 \\ 0 & - 1 \end{array} \right]\) State the equation of the line of invariant points. Circle your answer.
[0pt] [1 mark] $$x = 0 \quad y = 0 \quad y = x \quad y = - x$$

4. $$\mathbf { A } = \left( \begin{array} { c c } - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ - \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \end{array} \right)$$

1. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A }\).
2. Hence find the smallest positive integer value of \(n\) for which $$\mathbf { A } ^ { n } = \mathbf { I }$$ where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix. The transformation represented by the matrix \(\mathbf { A }\) followed by the transformation represented by the matrix \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\). Given that \(\mathbf { C } = \left( \begin{array} { r r } 2 & 4 \\ - 3 & - 5 \end{array} \right)\),
3. find the matrix \(\mathbf { B }\).

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

1. State the transformation represented by the matrix \(\left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right)\).
2. Write down the \(2 \times 2\) matrix for rotation through \(90 ^ { \circ }\) anticlockwise about the origin.
3. Find the \(2 \times 2\) matrix for rotation through \(90 ^ { \circ }\) anticlockwise about the origin, followed by reflection in the \(x\)-axis.

5. $$\mathbf { P } = \left( \begin{array} { r r } \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \\ \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } \end{array} \right)$$ The matrix \(\mathbf { P }\) represents the transformation \(U\)

1. Give a full description of \(U\) as a single geometrical transformation. The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the line \(y = - x\)
2. Write down the matrix \(\mathbf { Q }\) The transformation \(U\) followed by the transformation \(V\) is represented by the matrix \(\mathbf { R }\)
3. Determine the matrix \(\mathbf { R }\) The transformation \(W\) is represented by the matrix \(3 \mathbf { R }\) The transformation \(W\) maps a triangle \(T\) to a triangle \(T ^ { \prime }\) The transformation \(W ^ { \prime }\) maps the triangle \(T ^ { \prime }\) back to the original triangle \(T\)
4. Determine the matrix that represents \(W ^ { \prime }\)

7 A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\left( \begin{array} { c c } 1 & s \\ t & 0 \end{array} \right)\). Find in terms of \(s\) the matrices which represent each of the shears.

5. (a) \(\mathrm { P } , \mathrm { Q }\) and T are three transformations in 2-D. P is a reflection in the \(x\)-axis. A is the matrix that represents P . Write down the matrix A .
(b) \(Q\) is a shear in which the \(y\)-axis is invariant and the point \(\binom { 1 } { 0 }\) is transformed to the point \(\binom { 1 } { 2 }\). B is the matrix that represents Q . Find the matrix \(B\).
(c) T is P followed by Q. C is the matrix that represents T. Determine the matrix \(\mathbf { C }\).
(d) \(L\) is the line whose equation is \(y = x\). Explain whether or not \(L\) is a line of invariant points under \(T\).
(e) An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\). Explain what the value of the determinant of \(\mathbf { C }\) means about

• the area of \(N\) compared to the area of \(M\),
• the orientation of \(N\) compared to the orientation of \(M\).
  [0pt] [BLANK PAGE]

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } \cos 2 \theta & - \sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta \end{array} \right) \left( \begin{array} { l l } 1 & k \\ 0 & 1 \end{array} \right)\), where \(0 < \theta < \pi\) and \(k\) is a non-zero constant. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations, one of which is a shear.

1. Describe fully the other transformation and state the order in which the transformations are applied.
2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
3. Find, in terms of \(k\), the value of \(\tan \theta\) for which \(\mathbf { M - I }\) is singular.
4. Given that \(k = 2 \sqrt { 3 }\) and \(\theta = \frac { 1 } { 3 } \pi\), show that the invariant points of the transformation represented by \(\mathbf { M }\) lie on the line \(3 y + \sqrt { 3 } x = 0\).

1. (a) Prove by induction that for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ (b) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( r + 6 ) ( r - 6 ) = \frac { 1 } { 4 } n ( n + 1 ) ( n - 8 ) ( n + 9 )$$ (c) Hence find the value of \(n\) that satisfies $$\sum _ { r = 1 } ^ { n } r ( r + 6 ) ( r - 6 ) = 17 \sum _ { r = 1 } ^ { n } r ^ { 2 }$$

5. $$\mathbf { M } = \left( \begin{array} { c c } 1 & - \sqrt { 3 } \\ \sqrt { 3 } & 1 \end{array} \right)$$

1. Show that \(\mathbf { M }\) is non-singular. The hexagon \(R\) is transformed to the hexagon \(S\) by the transformation represented by the matrix \(\mathbf { M }\). Given that the area of hexagon \(R\) is 5 square units,
2. find the area of hexagon \(S\). The matrix \(\mathbf { M }\) represents an enlargement, with centre \(( 0,0 )\) and scale factor \(k\), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \(( 0,0 )\).
3. Find the value of \(k\).
4. Find the value of \(\theta\).

6.

1. $$\mathbf { B } = \left( \begin{array} { r r } - 1 & 2 \\ 3 & - 4 \end{array} \right) , \quad \mathbf { Y } = \left( \begin{array} { r r } 4 & - 2 \\ 1 & 0 \end{array} \right)$$ (a) Find \(\mathbf { B } ^ { - 1 }\). The transformation represented by \(\mathbf { Y }\) is equivalent to the transformation represented by \(\mathbf { B }\) followed by the transformation represented by the matrix \(\mathbf { A }\).
   (b) Find \(\mathbf { A }\).
2. $$\mathbf { M } = \left( \begin{array} { r r } - \sqrt { 3 } & - 1 \\ 1 & - \sqrt { 3 } \end{array} \right)$$ The matrix \(\mathbf { M }\) represents an enlargement scale factor \(k\), centre ( 0,0 ), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \(( 0,0 )\).
   (a) Find the value of \(k\).
   (b) Find the value of \(\theta\).

5 A transformation of the \(x - y\) plane is represented by the matrix \(\left( \begin{array} { r r } \cos \theta & 2 \sin \theta \\ 2 \sin \theta & - \cos \theta \end{array} \right)\), where \(\theta\) is a positive acute angle.

1. Write down the image of the point \(( 2,3 )\) under this transformation.
2. You are given that this image is the point ( \(a , 0\) ). Find the value of \(a\).

4. A right angled triangle \(T\) has vertices \(A ( 1,1 ) , B ( 2,1 )\) and \(C ( 2,4 )\). When \(T\) is transformed by the matrix \(\mathbf { P } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\), the image is \(T ^ { \prime }\).

1. Find the coordinates of the vertices of \(T ^ { \prime }\).
2. Describe fully the transformation represented by \(\mathbf { P }\). The matrices \(\mathbf { Q } = \left( \begin{array} { c c } 4 & - 2 \\ 3 & - 1 \end{array} \right)\) and \(\mathbf { R } = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)\) represent two transformations. When \(T\) is transformed by the matrix \(\mathbf { Q R }\), the image is \(T ^ { \prime \prime }\).
3. Find \(\mathbf { Q R }\).
4. Find the determinant of \(\mathbf { Q R }\).
5. Using your answer to part (d), find the area of \(T ^ { \prime \prime }\).

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.

8. The point \(( x , y , z )\) is rotated through \(60 ^ { \circ }\) anticlockwise around the \(z\)-axis. After rotation, the value of the \(x\)-coordinate is equal to the value of the \(y\)-coordinate.
Show that \(y = ( a + \sqrt { b } ) x\), where \(a\), \(b\) are integers whose values are to be determined.

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 \(\mathbf { M } ^ { 10 }\).

6 A transformation T is represented by the matrix \(\mathbf { T }\) where \(\mathbf { T } = \left( \begin{array} { c c } x ^ { 2 } + 1 & - 4 \\ 3 - 2 x ^ { 2 } & x ^ { 2 } + 5 \end{array} \right)\). A quadrilateral \(Q\), whose area is 12 units, is transformed by T to \(Q ^ { \prime }\). Find the smallest possible value of the area of \(Q ^ { \prime }\).

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 .

13
The transformation S is represented by the matrix \(\left[ \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right]\) The transformation T is a translation by the vector \(\left[ \begin{array} { c } 0 \\ - 5 \end{array} \right]\) Kamla transforms the graphs of various functions by applying first S , then T .
Leo says that, for some graphs, Kamla would get a different result if she applied first \(T\), then \(S\). Kamla disagrees.
State who is correct.
Fully justify your answer.

3 The matrix \(\mathbf { A } = \left[ \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right]\) represents a transformation.
Which one of the points below is an invariant point under this transformation?
Circle your answer. \(( 1,1 )\) \(( 0,2 )\) \(( 3,0 )\) \(( 2,1 )\)

4. The transformation \(T\) in the plane consists of a translation in which the point \(( x , y )\) is transformed to the point ( \(x + 2 , y - 2\) ), followed by a reflection in the line \(y = x\).

1. Determine the \(3 \times 3\) matrix which represents \(T\).
2. Determine how many invariant points exist under the transformation \(T\).

9

1. Write down the matrix \(\mathbf { C }\) which represents a stretch, scale factor 2 , in the \(x\)-direction.
2. The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { l l } 1 & 3 \\ 0 & 1 \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { D }\).
3. The matrix \(\mathbf { M }\) represents the combined effect of the transformation represented by \(\mathbf { C }\) followed by the transformation represented by \(\mathbf { D }\). Show that $$\mathbf { M } = \left( \begin{array} { l l } 2 & 3 \\ 0 & 1 \end{array} \right)$$
4. Prove by induction that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right) \\ 0 & 1 \end{array} \right)\), for all positive integers \(n\).

1. $$\mathbf { M } = \left( \begin{array} { l l } 4 & - 5 \\ 2 & - 7 \end{array} \right)$$

1. Show that the matrix \(\mathbf { M }\) is non-singular. The transformation \(T\) of the plane is represented by the matrix \(\mathbf { M }\).
   The triangle \(R\) is transformed to the triangle \(S\) by the transformation \(T\).
   Given that the area of \(S\) is 63 square units,
2. find the area of \(R\).
3. Show that the line \(y = 2 x\) is invariant under the transformation \(T\).

13. $$\mathbf { A } = \left( \begin{array} { c c } 2 & a \\ a - 4 & b \end{array} \right)$$ where \(a\) and \(b\) are non-zero constants.
Given that the matrix \(\mathbf { A }\) is self-inverse,

1. determine the value of \(b\) and the possible values for \(a\). The matrix \(\mathbf { A }\) represents a linear transformation \(M\).
   Using the smaller value of \(a\) from part (a),
2. show that the invariant points of the linear transformation \(M\) form a line, stating the equation of this line.
   [0pt] [BLANK PAGE]

2 A 2-D transformation \(T\) is a shear which leaves the \(y\)-axis invariant and which transforms the object point \(( 2,1 )\) to the image point \(( 2,9 )\). \(A\) is the matrix which represents the transformation \(T\).

1. Find A .
2. By considering the determinant of A , explain why the area of a shape is invariant under T .

12
The transformation S is represented by the matrix \(\mathbf { M } = \left[ \begin{array} { c c } 1 & - 6 \\ 2 & 7 \end{array} \right]\) The transformation T is a reflection in the line \(y = x \sqrt { 3 }\) and is represented by the matrix \(\mathbf { N }\) The point \(P ( x , y )\) is transformed first by S , then by T
The result of these transformations is the point \(Q ( 3,8 )\) Find the coordinates of \(P\) Give your answers to three decimal places.

8. $$\mathbf { P } = \left( \begin{array} { r r } 3 a & - 4 a \\ 4 a & 3 a \end{array} \right) , \text { where } a \text { is a constant and } a > 0$$

1. Find the matrix \(\mathbf { P } ^ { - 1 }\) in terms of \(a\).
   (3) The matrix \(\mathbf { P }\) represents the transformation \(U\) which transforms a triangle \(T _ { 1 }\) onto the triangle \(T _ { 2 }\).
   The triangle \(T _ { 2 }\) has vertices at the points ( \(- 3 a , - 4 a\) ), ( \(6 a , 8 a\) ), and ( \(- 20 a , 15 a\) ).
2. Find the coordinates of the vertices of \(T _ { 1 }\)
3. Hence, or otherwise, find the area of triangle \(T _ { 2 }\) in terms of \(a\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through an angle \(\alpha\) clockwise about the origin, where \(\tan \alpha = \frac { 4 } { 3 }\) and \(0 < \alpha < \frac { \pi } { 2 }\)
4. Write down the matrix \(\mathbf { Q }\), giving each element as an exact value. The transformation \(U\) followed by the transformation \(V\) is the transformation \(W\). The matrix \(\mathbf { R }\) represents the transformation \(W\).
5. Find the matrix \(\mathbf { R }\).

2. $$\mathbf { P } = \frac { 1 } { 2 } \left( \begin{array} { r r } 1 & \sqrt { 3 } \\ - \sqrt { 3 } & 1 \end{array} \right) \quad \mathbf { Q } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)$$ The matrices \(\mathbf { P }\) and \(\mathbf { Q }\) represent linear transformations, \(P\) and \(Q\) respectively, of the plane.
The linear transformation \(M\) is formed by first applying \(P\) and then applying \(Q\).

1. Find the matrix \(\mathbf { M }\) that represents the linear transformation \(M\).
2. Show that the invariant points of the linear transformation \(M\) form a line in the plane, stating the equation of this line.
   [0pt] [BLANK PAGE]