Standard Matrix Transformations (Edexcel International AS Further Maths: Further Pure 1): Revision Note

Reflection Matrices

How do I find reflection matrices?

• Imagine the unit square OABC

  • It has a side-length 1 unit

  • O is the origin

• The coordinates of A  and C as column vectors are

  • $A=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and $C=\left(\begin{array}{c}0\\ 1\end{array}\right)$

• Under a reflection about an axis or $y=\pm x$, A moves to A' and C moves to C'  

  • The matrix, $\mathbf{M}$ representing this reflection is $\mathbf{M}=\left(\begin{array}{cc}A\text{'}|& C\text{'}\end{array}\right)$

  • A' and C' are column vectors of their new positions

    • The points O  and B  are not needed, as we can draw the reflected square using just A' and C' (O  won't move)

• For example:

  • To find the matrix representing a reflection about the $x$-axis

    • A stays where it is, so $A\text{'}=\left(\begin{array}{c}1\\ 0\end{array}\right)$

    • C goes to $C\text{'}=\left(\begin{array}{c}0\\ -1\end{array}\right)$ (on the negative $y$-axis)

    • $\mathbf{M}=\left(\begin{array}{cc}A\text{'}|& C\text{'}\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$

  • To find the matrix representing a reflection in the line $y=x$

    • A goes to $A\text{'}=\left(\begin{array}{c}0\\ 1\end{array}\right)$ (on the positive y-axis)

    •  C goes to $C\text{'}=\left(\begin{array}{c}1\\ 0\end{array}\right)$ (on the positive x-axis)

    • $\mathbf{M}=\left(\begin{array}{cc}A\text{'}|& C\text{'}\end{array}\right)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$

    • (This is not the same as the identity matrix, as the 1s are on the wrong diagonal)

Enlargement & Stretch Matrices

Which matrix represents an enlargement?

• The matrix $\mathbf{M}=\left(\begin{array}{cc}k& 0\\ 0& k\end{array}\right)$ represents an enlargement of scale factor $k$ about the origin, O
  

  • This is the same as $\mathbf{M}=k\mathbf{I}$

    • $\mathbf{I}$ is the identity matrix

Which matrix represents a stretch?

• The matrix $\mathbf{M}=\left(\begin{array}{cc}a& 0\\ 0& 1\end{array}\right)$ represents a stretch parallel to the $x$-axis of scale factor $a$

  • The point $\left(x,y\right)$ becomes $\left(ax,y\right)$

• The matrix $\mathbf{M}=\left(\begin{array}{cc}1& 0\\ 0& b\end{array}\right)$ represents a stretch parallel to the $y$-axis of scale factor $b$

  • The point $\left(x,y\right)$ becomes $\left(x,by\right)$

• The matrix $\mathbf{M}=\left(\begin{array}{cc}a& 0\\ 0& b\end{array}\right)$ represents a combined stretch of scale factor $a$ parallel to the $x$-axis and scale factor $b$ parallel to the $y$-axis

  • If $a=b$, the combined stretch is an enlargement

Rotation Matrices

How do I find matrices for rotations by multiples of 90°?

• Imagine the unit square OABC

  • It has a side-length of 1 unit

  • O is the origin

• The coordinates of A and C as column vectors are

  • $A=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and $C=\left(\begin{array}{c}0\\ 1\end{array}\right)$

• Under a rotation about the origin, A moves to A' and C moves to C' 

  • The matrix, $\mathbf{M}$ representing this rotation is $\mathbf{M}=\left(\begin{array}{cc}A\text{'}|& C\text{'}\end{array}\right)$

  • A' and C' are column vectors of their new positions

    • The points O and B are not needed, as we can draw the rotated square using just A' and C' (O won't move)

• For example:

  • To find the matrix representing a rotation of 90° anticlockwise about the origin

    • A goes to $A\text{'}=\left(\begin{array}{c}0\\ 1\end{array}\right)$ (on the positive $y$-axis)

    • C goes to $C\text{'}=\left(\begin{array}{c}-1\\ 0\end{array}\right)$ (on the negative $x$-axis)

    • $\mathbf{M}=\left(\begin{array}{cc}A\text{'}|& C\text{'}\end{array}\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$

  • To find the matrix representing a rotation of 180° about the origin

    • A goes to $A\text{'}=\left(\begin{array}{c}-1\\ 0\end{array}\right)$ (on the negative $x$-axis)

    • C goes to $C\text{'}=\left(\begin{array}{c}0\\ -1\end{array}\right)$ (on the negative $y$-axis)

    • $\mathbf{M}=\left(\begin{array}{cc}A\text{'}|& C\text{'}\end{array}\right)=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)$

    • This is the same as $\mathbf{M}=-\mathbf{I}$ where $\mathbf{I}$ is the identity matrix

How do I find matrices for rotations by any angle?

• A rotation anticlockwise by any angle, $\theta$, about the origin is represented by the matrix:

  • $\mathbf{M}=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$

    • $\theta$ can be in degrees or radians

• A negative value of $\theta$ represents a clockwise rotation

  • Remember that $\sin \left(-\theta \right)=-\sin \theta$ but that $\cos \left(-\theta \right)=\cos \theta$

• You can substitute in multiples of 90° to get the matrices above

• You may be required to recognise common angles from their ratios

  • For example, $\frac{\sqrt{3}}{2}=\cos \left(\frac{\mathrm{\pi }}{6}\right)$