Linear transformations Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Linear transformations questions. See exactly how to solve problems on linear-transformations, rotation, matrix, reflection.

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Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
Write down the matrix that represents a rotation of 90∘90^\circ anticlockwise about the origin.

Worked solution

  1. Write down the general rotation matrix

    (cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

    An anticlockwise rotation through θ\theta has this matrix.

  2. Substitute θ=90∘\theta=90^\circ

    cos⁡90∘=0,sin⁡90∘=1\cos 90^\circ=0,\quad \sin 90^\circ=1

    Evaluate the trigonometric functions exactly.

  3. State the rotation matrix

    (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}

    This matrix rotates every point about the origin.

Answer
(0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}
Question 2
2 markseasy
Write down the matrix that represents a rotation of 180∘180^\circ anticlockwise about the origin.

Worked solution

  1. Write down the general rotation matrix

    (cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

    An anticlockwise rotation through θ\theta has this matrix.

  2. Substitute θ=180∘\theta=180^\circ

    cos⁡180∘=−1,sin⁡180∘=0\cos 180^\circ=-1,\quad \sin 180^\circ=0

    Evaluate the trigonometric functions exactly.

  3. State the rotation matrix

    (−100−1)\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}

    This matrix rotates every point about the origin.

Answer
(−100−1)\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}
Question 3
2 markseasy
Write down the matrix that represents a rotation of 270∘270^\circ anticlockwise about the origin.

Worked solution

  1. Write down the general rotation matrix

    (cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

    An anticlockwise rotation through θ\theta has this matrix.

  2. Substitute θ=270∘\theta=270^\circ

    cos⁡270∘=0,sin⁡270∘=−1\cos 270^\circ=0,\quad \sin 270^\circ=-1

    Evaluate the trigonometric functions exactly.

  3. State the rotation matrix

    (01−10)\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}

    This matrix rotates every point about the origin.

Answer
(01−10)\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}
Question 4
2 markseasy
Write down the matrix that represents a reflection in the xx-axis.

Worked solution

  1. Write down the general reflection matrix

    (cos⁡2θsin⁡2θsin⁡2θ−cos⁡2θ)\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}

    Here θ\theta is the angle between the mirror line and the positive xx-axis.

  2. Identify the angle of the mirror line

    θ=0∘ ⇒ 2θ=0∘\theta=0^\circ\ \Rightarrow\ 2\theta=0^\circ

    The mirror line makes an angle of 0∘0^\circ with the positive xx-axis.

  3. State the reflection matrix

    (100−1)\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

    This matrix reflects every point in the given line.

Answer
(100−1)\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
Question 5
2 markseasy
Write down the matrix that represents a reflection in the yy-axis.

Worked solution

  1. Write down the general reflection matrix

    (cos⁡2θsin⁡2θsin⁡2θ−cos⁡2θ)\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}

    Here θ\theta is the angle between the mirror line and the positive xx-axis.

  2. Identify the angle of the mirror line

    θ=90∘ ⇒ 2θ=180∘\theta=90^\circ\ \Rightarrow\ 2\theta=180^\circ

    The mirror line makes an angle of 90∘90^\circ with the positive xx-axis.

  3. State the reflection matrix

    (−1001)\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}

    This matrix reflects every point in the given line.

Answer
(−1001)\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}

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