Find the standard matrix for the reflection of R^2 about the...

Find the standard matrix for the reflection of $R^{2}$ about the stated line, and then use that matrix to find the reflection of the given point about that line. The reflection of (1,2) about the line that makes an angle of $\pi / 4\left(=45^{\circ}\right)$ with the positive $x$ -axis.

Find the standard matrix for the reflection of $R^{2}$ about the stated line, and then use that matrix to find the reflection of the given point about that line.
The reflection of (1,2) about the line that makes an angle of $\pi / 4\left(=45^{\circ}\right)$ with the positive $x$ -axis.

Key Concepts

Reflection Transformation

A reflection transformation in Euclidean space is an isometry that 'flips' points over a specified line (or plane in higher dimensions), producing a mirror image. In R², reflecting a point about a line reverses the perpendicular distance to that line while keeping the parallel component unchanged.

Standard Matrix Representation

Any linear transformation in R² can be represented by a 2×2 matrix called the standard matrix. The columns of this matrix are the images of the standard basis vectors under the transformation, which allows the transformation to be applied to any vector through matrix multiplication.

Angle-based Reflection Matrix

When reflecting points about a line that passes through the origin and makes an angle ? with the positive x-axis, the standard matrix for the reflection is given by [[cos 2?, sin 2?], [sin 2?, -cos 2?]]. This formulation shows how the reflection depends on the double angle, effectively encoding how the geometry of the reflection operates relative to the specified line orientation.

Matrix Multiplication for Transformation

After determining the appropriate reflection matrix, the reflected image of any point in R² can be found by multiplying the reflection matrix by the column vector representing the point’s coordinates. This process demonstrates the application of the transformation to each point in the plane.

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