Let L ⊂ℝ^2 be the line through the origin in the direction...

Let $L \subset \mathbb{R}^2$ be the line through the origin in the direction of a unit vector $\mathbf{u}$. (a) Prove that the matrix representative of reflection through $L$ is $R=2 \mathbf{u u}^T-1$. (b) Find the corresponding formula for reflection through the line in the direction of a general nonzero vector $\mathbf{v} \neq \mathbf{0}$. (c) Determine the matrix representative for reflection through the line in the direction $(i)(1,0)^T$, (ii) $\left(\frac{3}{5},-\frac{4}{5}\right)^T$, (iii) $(1,1)^T$, (iv) $(2,-3)^T$.

Let $L \subset \mathbb{R}^2$ be the line through the origin in the direction of a unit vector $\mathbf{u}$. (a) Prove that the matrix representative of reflection through $L$ is $R=2 \mathbf{u u}^T-1$. (b) Find the corresponding formula for reflection through the line in the direction of a general nonzero vector $\mathbf{v} \neq \mathbf{0}$. (c) Determine the matrix representative for reflection through the line in the direction $(i)(1,0)^T$,
(ii) $\left(\frac{3}{5},-\frac{4}{5}\right)^T$,
(iii) $(1,1)^T$,
(iv) $(2,-3)^T$.

Key Concepts

Unit Vector Normalization

Normalization is the process of converting any nonzero vector into a unit vector—one with a magnitude of 1—while maintaining its direction. This step is essential when using the formula R = 2uu^T - I for reflection, as the validity of this formula depends on the vector u being of unit length. If the direction is specified by a non-unit vector, normalization must be performed first to correctly form the reflection matrix.

Generalization to Non-Unit Vectors

When the line of reflection is provided by a nonzero, non-unit vector, the reflection formula must be adapted to account for the vector’s magnitude. This involves normalizing the vector to obtain the appropriate unit vector representation before applying the reflection formula. Such generalization ensures that the reflection matrix accurately represents the transformation regardless of the initial scale of the direction vector.

Reflection Transformation

Reflection is a linear transformation that mirrors each point in a space across a given line (or hyperplane in higher dimensions). In the context of ?², reflecting a vector through a line means flipping its component perpendicular to the line while leaving the component parallel to the line unchanged. This transformation is represented by a matrix that has specific properties, including being orthogonal and having a determinant of -1.

Projection onto a Line

Projection involves decomposing a vector into two components: one that is parallel to a given line and one that is perpendicular. The projection of a vector onto a line given by a unit vector u is achieved using the dot product, resulting in a new vector that lies along the line. In deriving the reflection matrix, this concept is crucial because the reflection operation essentially doubles the projection (the parallel component) and then subtracts the original vector to reverse the perpendicular component.

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