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

$$\mathbf{P} = \begin{pmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}$$ The matrix \(\mathbf{P}\) represents the transformation \(U\)

1. Give a full description of \(U\) as a single geometrical transformation. [2]

The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf{Q}\), is a reflection in the line \(y = -x\)

1. Write down the matrix \(\mathbf{Q}\) [1]

The transformation \(U\) followed by the transformation \(V\) is represented by the matrix \(\mathbf{R}\)

1. Determine the matrix \(\mathbf{R}\) [2]

The transformation \(W\) is represented by the matrix \(3\mathbf{R}\) The transformation \(W\) maps a triangle \(T\) to a triangle \(T'\) The transformation \(W'\) maps the triangle \(T'\) back to the original triangle \(T\)

1. Determine the matrix that represents \(W'\) [3]

$$\mathbf{A} = \begin{pmatrix} 3\sqrt{2} & 0 \\ 0 & 3\sqrt{2} \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$

1. Describe fully the transformations described by each of the matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\). [4]

It is given that the matrix \(\mathbf{D} = \mathbf{CA}\), and that the matrix \(\mathbf{E} = \mathbf{DB}\).

1. Find \(\mathbf{D}\). [2]
2. Show that \(\mathbf{E} = \begin{pmatrix} -3 & 3 \\ 3 & 3 \end{pmatrix}\). [1]

The triangle \(ORS\) has vertices at the points with coordinates \((0, 0)\), \((-15, 15)\) and \((4, 21)\). This triangle is transformed onto the triangle \(OR'S'\) by the transformation described by \(\mathbf{E}\).

1. Find the coordinates of the vertices of triangle \(OR'S'\). [4]
2. Find the area of triangle \(OR'S'\) and deduce the area of triangle \(ORS\). [3]

A transformation \(T: \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix $$\mathbf{A} = \begin{pmatrix} -4 & 2 \\ 2 & -1 \end{pmatrix}, \text{ where } k \text{ is a constant.}$$ Find the image under \(T\) of the line with equation \(y = 2x + 1\). [2]

The matrix \(\mathbf{A}\) is defined by \(\mathbf{A} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\).

1. 1. Find the matrix \(\mathbf{A}^2\). [1 mark]
   2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf{A}^2\). [1 mark]
2. Given that the matrix \(\mathbf{B}\) represents a reflection in the line \(x + \sqrt{3}y = 0\), find the matrix \(\mathbf{B}\), giving the exact values of any trigonometric expressions. [2 marks]
3. Hence find the coordinates of the point \(P\) which is mapped onto \((0, -4)\) under the transformation represented by \(\mathbf{A}^2\) followed by a reflection in the line \(x + \sqrt{3}y = 0\). [6 marks]

\includegraphics{figure_6} The diagram shows the unit square \(OABC\), and its image \(OA'B'C'\) after a transformation. The points have the following coordinates: \(A(1, 0)\), \(B(1, 1)\), \(C(0, 1)\), \(B'(3, 2)\) and \(C'(2, 2)\).

1. Write down the matrix, X, for this transformation. [2]
2. The transformation represented by X is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a pair of possible transformations P and Q and state the matrices that represent them. [6]
3. Find the matrix that represents transformation Q followed by transformation P. [2]

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]

The transformation S is represented by the matrix \(\begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}\) The transformation T is a translation by the vector \(\begin{pmatrix} 0 \\ -5 \end{pmatrix}\) 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 marks]

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]

The transformation \(P\) is an enlargement, centre the origin, with scale factor \(k\), where \(k > 0\) The transformation \(Q\) is a rotation through angle \(\theta\) degrees anticlockwise about the origin. The transformation \(P\) followed by the transformation \(Q\) is represented by the matrix $$\mathbf{M} = \begin{pmatrix} -4 & -4\sqrt{3} \\ 4\sqrt{3} & -4 \end{pmatrix}$$

1. Determine
   1. the value of \(k\),
   2. the smallest value of \(\theta\)
   [4] A square \(S\) has vertices at the points with coordinates \((0, 0)\), \((a, -a)\), \((2a, 0)\) and \((a, a)\) where \(a\) is a constant. The square \(S\) is transformed to the square \(S'\) by the transformation represented by \(\mathbf{M}\).
2. Determine, in terms of \(a\), the area of \(S'\) [2]

P, Q and T are three transformations in 2-D. P is a reflection in the \(x\)-axis. A is the matrix that represents P.

1. Write down the matrix A. [1]

Q is a shear in which the \(y\)-axis is invariant and the point \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) is transformed to the point \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\). B is the matrix that represents Q.

1. Find the matrix B. [2]

T is P followed by Q. C is the matrix that represents T.

1. Determine the matrix C. [2]

\(L\) is the line whose equation is \(y = x\).

1. Explain whether or not \(L\) is a line of invariant points under T. [2]

An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\).

1. Explain what the value of the determinant of C means about
   [3]

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]

Transformation M is represented by matrix \(\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\).

1. On the diagram in the Printed Answer Booklet draw the image of the unit square under M. [2]
2. 1. Show that there is a constant \(k\) such that \(\mathbf{M} \begin{pmatrix} x \\ kx \end{pmatrix} = 5 \begin{pmatrix} x \\ kx \end{pmatrix}\) for all \(x\). [2]
   2. Hence find the equation of an invariant line under M. [1]
   3. Draw the invariant line from part (ii) (B) on your diagram for part (i). [1]

The matrix A is \(\begin{pmatrix} 0.6 & 0.8 \\ 0.8 & -0.6 \end{pmatrix}\)

1. Given that A represents a reflection, write down the eigenvalues of A. [1]
2. Hence find the eigenvectors of A. [3]
3. Write down the equation of the mirror line of the reflection represented by A. [1]

The transformation \(T\) in the plane consists of a translation in which the point \((x, y)\) is transformed to the point \((x - 1, y + 1)\), followed by a reflection in the line \(y = x\).

1. Determine the \(3 \times 3\) matrix which represents \(T\). [4]
2. Find the equation of the line of fixed points of \(T\). [2]
3. Find \(T^2\) and hence write down \(T^{-1}\). [3]

The transformation \(T\) in the plane consists of a reflection in the line \(y = x\), followed by a translation in which the point \((x, y)\) is transformed to the point \((x + 1, y - 2)\), followed by an anticlockwise rotation through \(90°\) about the origin.

1. Find the \(3 \times 3\) matrix representing \(T\). [6]
2. Show that \(T\) has no fixed points. [3]

The \(2 \times 2\) matrix A represents a rotation by \(90°\) anticlockwise about the origin. The \(2 \times 2\) matrix B represents a reflection in the line \(y = -x\). The matrix B is given by $$\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$$

1. Write down the matrix representing A. [1]
2. The \(2 \times 2\) matrix C represents a rotation by \(90°\) anticlockwise about the origin, followed by a reflection in the line \(y = -x\). Compute the matrix C and describe geometrically the single transformation represented by C. [3]

The \(2 \times 2\) matrix \(\mathbf{A}\) 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 \(\mathbf{A}\). [8]

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

1. Find the equation of the line of invariant points of \(S\). [2]
2. Show that any line of the form \(y = x + c\) is an invariant line of \(S\). [3]

You are given the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}\).

1. Find \(\mathbf{A}^4\). [1]
2. Describe the transformation that \(\mathbf{A}\) represents. [2]

The matrix \(\mathbf{B}\) represents a reflection in the plane \(x = 0\).

1. Write down the matrix \(\mathbf{B}\). [1]

The point \(P\) has coordinates \((2, 3, 4)\). The point \(P'\) is the image of \(P\) under the transformation represented by \(\mathbf{B}\).

1. Find the coordinates of \(P'\). [1]