The question is:
Determine the matrix of the following reflection in $\Bbb R^2$: $R$ is a relection in the line $x_1-5x_2 = 0$.
The question is:
Determine the matrix of the following reflection in $\Bbb R^2$: $R$ is a relection in the line $x_1-5x_2 = 0$.
$ \begin{bmatrix} 0.9231 & 0.3846 \\ 0.3846 & -0.9231 \end{bmatrix} $ You can evaluate this easily by looking at a simple reflection in the first place. Say about the $1^{st}$ axis (corresponding to $x_{1}$). This can be represented by $ \mathbf{I}_{ref} = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} $ Now you can think of a general reflection about a line $x_{2} = tan(\theta)x_{1}$. Rotate the point in question by $-\theta$ about the origin (rotation matrix $\mathbf{R}_{-\theta}$). Now the reflection would be about the first axis ($\mathbf{I}_{ref}$). Now just rotate it back by $\theta$. The required reflection matrix is given by $ \mathbf{R}_{\theta} \mathbf{I}_{ref} \mathbf{R}_{-\theta} $