Linear transformations

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. State the transformation represented by the matrix \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\). [1]
2. Write down the \(2 \times 2\) matrix for rotation through \(90°\) anticlockwise about the origin. [1]
3. Find the \(2 \times 2\) matrix for rotation through \(90°\) anticlockwise about the origin, followed by reflection in the \(x\)-axis. [2]

9 The plane transformation \(T\) is the composition (in this order) of

• a reflection in the line \(y = x \tan \frac { 1 } { 8 } \pi\); followed by
• a shear parallel to the \(y\)-axis, mapping \(( 1,0 )\) to \(( 1,2 )\); followed by
• a clockwise rotation through \(\frac { 1 } { 4 } \pi\) radians about the origin; followed by
• a shear parallel to the \(x\)-axis, mapping \(( 0,1 )\) to \(( - 2,1 )\).

Determine the matrix \(\mathbf { M }\) which represents \(T\), and hence give a full geometrical description of \(T\) as a single plane transformation.

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 \(\begin{pmatrix} 1 & s \\ t & 0 \end{pmatrix}\). Find in terms of s the matrices which represent each of the shears. [7]

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 .

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 \(3 y + \sqrt { 3 } x = 0\).

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 .

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 .

1. Write down the matrix \(\mathbf{C}\) which represents a stretch, scale factor \(2\), in the \(x\)-direction. [2]
2. The matrix \(\mathbf{D}\) is given by \(\mathbf{D} = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}\). Describe fully the geometrical transformation represented by \(\mathbf{D}\). [2]
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 $$\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$$ [2]
4. Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}\), for all positive integers \(n\). [6]

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 \(\theta\).

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 0.6 & 2.4 \\ -0.8 & 1.8 \end{pmatrix}\).

1. Find \(\det \mathbf{A}\). [1]

The matrix \(\mathbf{A}\) represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.

1. By considering the determinants of these transformations, determine the scale factor of the stretch. [2]
2. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer. [1]
3. Find the angle of rotation. [2]

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 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 \(R\).

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

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

1. Write down the image of the point \((2, 3)\) under this transformation. [2]
2. You are given that this image is the point \((a, 0)\). Find the value of \(a\). [5]

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.

A transformation T is represented by the matrix \(\mathbf{T}\) where \(\mathbf{T} = \begin{pmatrix} x^2 + 1 & -4 \\ 3 - 2x^2 & x^2 + 5 \end{pmatrix}\). A quadrilateral \(Q\), whose area is 12 units, is transformed by T to \(Q'\). Find the smallest possible value of the area of \(Q'\). [5]

1 You are given that matrix \(\mathbf { A } = \left( \begin{array} { r r } 2 & - 1 \\ 0 & 3 \end{array} \right)\) and matrix \(\mathbf { B } = \left( \begin{array} { r r } 3 & 1 \\ - 2 & 4 \end{array} \right)\).

1. Find BA.
2. A plane shape of area 3 square units is transformed using matrix \(\mathbf { A }\). The image is transformed using matrix B. What is the area of the resulting shape?

16. $$\begin{gathered} M _ { 1 } = \left( \begin{array} { c c } 2 k - 9 & 5 - k \\ - k & k - 2 \end{array} \right) \\ M _ { 2 } = \left( \begin{array} { c c } 5 & 1 \\ 2 k - 3 & k - 3 \end{array} \right) \\ k \in \mathbb { R } \end{gathered}$$ Matrices \(M _ { 1 }\) and \(M _ { 2 }\) represent transformations \(T _ { 1 }\) and \(T _ { 2 }\) respectively. \(\Delta\) is a triangle in the \(x y\)-plane with vertices at \(( 0,0 ) , ( 4,0 )\) and \(( 3,2 )\).
The image of \(\Delta\) under \(T _ { 1 }\) is \(\Delta _ { 1 }\) and the image of \(\Delta\) under \(T _ { 2 }\) is \(\Delta _ { 2 }\).
The area of \(\Delta _ { 2 }\) is greater than the area of \(\Delta _ { 1 }\).
Find the range of possible values of \(k\).
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6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$ Describe fully the geometrical transformation represented by each of the following matrices:

1. A ;
2. B ;
3. \(\quad \mathbf { A } ^ { 2 }\);
4. \(\quad \mathbf { B } ^ { 2 }\);
5. AB.
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1. $$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right) \quad \mathbf { Q } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 3 \end{array} \right)$$

1. 1. Describe fully the single geometrical transformation \(P\) represented by the matrix \(\mathbf { P }\).
   2. Describe fully the single geometrical transformation \(Q\) represented by the matrix \(\mathbf { Q }\). The transformation \(P\) followed by the transformation \(Q\) is the transformation \(R\), which is represented by the matrix \(\mathbf { R }\).
2. Determine \(\mathbf { R }\).
3. 1. Evaluate the determinant of \(\mathbf { R }\).
   2. Explain how the value obtained in (c)(i) relates to the transformation \(R\).

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 \(\mathbf { B }\).

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.

2. $$\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 5 & 3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } - 3 & - 1 \\ 5 & 2 \end{array} \right)$$

1. Find \(\mathbf { A B }\). Given that $$\mathbf { C } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)$$
2. describe fully the geometrical transformation represented by \(\mathbf { C }\),
3. write down \(\mathbf { C } ^ { 100 }\). \includegraphics[max width=\textwidth, alt={}, center]{d20fa710-2d91-4ac2-adbc-46ccdcb93380-03_99_97_2631_1784}

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 .

The transformation T is defined by the matrix M. The transformation S is defined by the matrix \(\mathbf{M}^{-1}\). Given that the point \((x, y)\) is invariant under transformation T, prove that \((x, y)\) is also an invariant point under transformation S. [3 marks]

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 \(T\).

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. 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 \(T\).

6 The matrices \(\mathbf { M }\) and \(\mathbf { N }\) are \(\left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\) respectively.

1. In this question you must show detailed reasoning. Determine whether \(\mathbf { M }\) and \(\mathbf { N }\) commute under matrix multiplication.
2. Specify the transformation of the plane associated with each of the following matrices.
   1. M
   2. N
3. State the significance of the result in part (a) for the transformations associated with \(\mathbf { M }\) and \(\mathbf { N }\). [1]
4. Use an algebraic method to show that all lines parallel to the \(x\)-axis are invariant lines of the transformation associated with N.

Describe fully the transformation given by the matrix \(\begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix}\) [3 marks]

Three non-singular square matrices, \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{R}\) are such that $$\mathbf{AR} = \mathbf{B}$$ The matrix \(\mathbf{R}\) represents a rotation about the \(z\)-axis through an angle \(\theta\) and $$\mathbf{B} = \begin{pmatrix} -\cos \theta & \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

1. Show that \(\mathbf{A}\) is independent of the value of \(\theta\). [3 marks]
2. Give a full description of the single transformation represented by the matrix \(\mathbf{A}\). [1 mark]

3. (a) Given that $$\mathbf { A } = \left( \begin{array} { c c } 1 & \sqrt { } 2 \\ \sqrt { } 2 & - 1 \end{array} \right)$$

1. find \(\mathbf { A } ^ { 2 }\),
2. describe fully the geometrical transformation represented by \(\mathbf { A } ^ { 2 }\).
   (b) Given that $$\mathbf { B } = \left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)$$ describe fully the geometrical transformation represented by \(\mathbf { B }\).
   (c) Given that $$\mathbf { C } = \left( \begin{array} { c c } k + 1 & 12 \\ k & 9 \end{array} \right)$$ where \(k\) is a constant, find the value of \(k\) for which the matrix \(\mathbf { C }\) is singular.

The transformation \(S\) is represented by the matrix \(\mathbf{M} = \begin{bmatrix} 1 & -6 \\ 2 & 7 \end{bmatrix}\) 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. [5 marks]

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 .

$$\mathbf { A } = \left( \begin{array} { c c } 2 & a \\ a - 4 & b \end{array} \right)$$ 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.
   (ii) $$\mathbf { P } = \left( \begin{array} { c c } p & 2 p \\ - 1 & 3 p \end{array} \right)$$ where \(p\) is a positive constant.
   The matrix \(\mathbf { P }\) represents a linear transformation \(U\).
   The triangle \(T\) has vertices at the points with coordinates ( 1,2 ), ( 3,2 ) and ( 2,5 ). The area of the image of \(T\) under the linear transformation \(U\) is 15
3. Determine the value of \(p\). The transformation \(V\) consists of a stretch scale factor 3 parallel to the \(x\)-axis with the \(y\)-axis invariant followed by a stretch scale factor - 2 parallel to the \(y\)-axis with the \(x\)-axis invariant. The transformation \(V\) is represented by the matrix \(\mathbf { Q }\).
4. Write down the matrix \(\mathbf { Q }\). Given that \(U\) followed by \(V\) is the transformation \(W\), which is represented by the matrix \(\mathbf { R }\), (c) find the matrix \(\mathbf { R }\).

1. Describe fully the transformation represented by the matrix \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\). [2]
2. A triangle of area 5 square units undergoes the transformation represented by the matrix \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\). Explaining your reasoning, find the area of the image of the triangle following this transformation. [2]