Question

Show that the Transformation matrix of reflection about line y = x is equivalent to reflection relative to x-axis followed by anticlockwise rotation of 90 degree

Answer 1

The transformation matrix of reflection about line y = x is equivalent to reflection relative to x-axis followed by anticlockwise rotation of 90 degrees because their transformation matrices are equal.

Answer 2

To show the equivalence of transformations, we determine the transformation matrices for each operation and then perform matrix multiplication.

1. Transformation Matrix for Reflection about the line $y = x$

   A point $(x, y)$ reflected about the line $y = x$ transforms to $(y, x)$. The transformation equations are: $x' = y$ $y' = x$ In matrix form, this is:

   $$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$

   So, the transformation matrix for reflection about $y = x$ is $M_{y=x} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.

2. Transformation Matrix for Reflection relative to the x-axis

   A point $(x, y)$ reflected about the x-axis transforms to $(x, -y)$. The transformation equations are: $x' = x$ $y' = -y$ In matrix form, this is:

   $$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$

   So, the transformation matrix for reflection about the x-axis is $M_x = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.

3. Transformation Matrix for Anticlockwise Rotation of 90 degrees

   The general transformation matrix for an anticlockwise rotation by an angle $\theta$ is:

   $$R_{\theta} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$

   For an anticlockwise rotation of $90^\circ$ ($\theta = 90^\circ$): $\cos 90^\circ = 0$ $\sin 90^\circ = 1$

   So, the transformation matrix for $90^\circ$ anticlockwise rotation is $R_{90} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.

4. Combined Transformation Matrix (Reflection relative to x-axis followed by Anticlockwise Rotation of 90 degrees)

   When transformations are applied sequentially, the matrices are multiplied in the reverse order of application. If transformation A is applied first, then B, the combined matrix is $M_B M_A$. Here, reflection about the x-axis ($M_x$) is applied first, followed by anticlockwise rotation of $90^\circ$ ($R_{90}$). The combined transformation matrix is $R_{90} \cdot M_x$:

   $$R_{90} \cdot M_x = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

   Performing the matrix multiplication:

   $$R_{90} \cdot M_x = \begin{pmatrix} (0)(1) + (-1)(0) & (0)(0) + (-1)(-1) \\ (1)(1) + (0)(0) & (1)(0) + (0)(-1) \end{pmatrix}$$ $$R_{90} \cdot M_x = \begin{pmatrix} 0+0 & 0+1 \\ 1+0 & 0+0 \end{pmatrix}$$ $$R_{90} \cdot M_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$

Conclusion

Comparing the combined transformation matrix $R_{90} \cdot M_x$ with the matrix for reflection about $y=x$:

$$R_{90} \cdot M_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$ $$M_{y=x} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$

Since $R_{90} \cdot M_x = M_{y=x}$, it is shown that the transformation matrix of reflection about the line $y = x$ is equivalent to reflection relative to the x-axis followed by an anticlockwise rotation of 90 degrees.