Determine the matrix of the reflection over the plane $2 x_1 + x_2 − 2 x_3 = 0$ in $\Bbb R^3$

Question

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Determine the matrix of the reflection over the plane $2x_1 +x_2 −2x_3 = 0 \in \mathbb{ R^3}$ .

Answer 1

Hint:

1. Find projection matrix $P$ (by projecting bases of $\mathbb R^3$ over the given plane.)
   
2. Hence, projection of a vector $b$ onto the plane will be $Pb$. Let the reflection of $b$ over the plane be $x$.
3. Now you know that, foot of projection is the middle point of line joining $b$ and $x$ and therefore, $Pb=(b+x) /2\implies (2P-I_3)b=x$, where $I_3$ is the identity matrix of order $3\times 3$
   Reflection matrix is $2P-I_3$ because it transforms any vector(b) in $\mathbb R^3$ into its reflection over the given plane.