Question

Again assume you are working in two, rather than three, dimensions. Determine the four entries of the \(2 \times 2\) reflection matrix that takes a vertex point at position \((x, y)\) and reflects it around the \(y\)-axis. That is, assume the mirror line in Figure \(16.6(c)\) is the \(y\)-axis. This reflection operation is shown here:

Short answer

The reflection matrix is \( \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \).

Step-by-step solution

Understanding Reflection Across the y-axis

When reflecting a point \( (x, y) \) across the y-axis, only the x-coordinate changes, while the y-coordinate remains the same. Specifically, if a point is \( (x, y) \), it will reflect to \( (-x, y) \).

Reflection Matrix Definition

A reflection matrix is a transformation matrix that, when multiplied by a position vector, produces the reflected vector. For reflection across the y-axis, the matrix should transform \( (x,y) \) to \( (-x,y) \).

Formulate the Reflection Matrix

To create the reflection matrix that reflects a vector \( (x, y) \) into \( (-x, y) \), the matrix must be \\[\begin{bmatrix}-1 & 0 \0 & 1 \\end{bmatrix}\].This matrix will transform any point \( (x,y) \) to \( (-x, y) \).

Verify the Reflection Matrix

Verify the transformation by multiplying the matrix with a position vector. Take a vector \( (x, y) \) and multiply it with the matrix: \\[\begin{bmatrix}-1 & 0 \0 & 1 \\end{bmatrix} \begin{bmatrix}x \y \\end{bmatrix} = \begin{bmatrix}-x \y \\end{bmatrix}\].This gives the reflected vector \( (-x, y) \), validating the reflection matrix.

Key concepts

2x2 Matrix

A 2x2 matrix is a type of matrix used in linear algebra. It has two rows and two columns, making four entries in total. These matrices are particularly useful when performing transformations in two-dimensional space, such as reflections, rotations, or scaling.
This format of a matrix is generally represented as: \[\begin{bmatrix}a & b \c & d \end{bmatrix}\] where \(a, b, c,\) and \(d\) are the elements of the matrix. This simple structure makes them ideal for transforming coordinates in two-dimensional geometry.

• Each entry can be manipulated to achieve different transformation effects.
• They apply transformations directly to coordinate points through matrix multiplication.
• Common 2x2 matrices include identity matrices, rotation matrices, and reflection matrices.

Understanding 2x2 matrices is fundamental in performing and visualizing geometric transformations in a plane, as they succinctly encode operations like reflections that change object positions accordingly.

Linear Transformation

Linear transformations are functions between vector spaces that preserve vector addition and scalar multiplication. In simpler terms, they maintain the origin and straight lines, fully characterized by matrices. In the case of a 2D space, a 2x2 matrix describes these transformations.
For instance, when transforming a point \((x, y)\), the action of linear transformation can be visualized by the application of a matrix to a vector:

• Reflection: A common linear transformation. When reflected across the y-axis, effects include changes to only the x-coordinates of points.
• The transformation is expressed using a reflection matrix, generally taking a form where one axis is negated to achieve a mirror image.

In reflecting across the y-axis, the matrix impacts vectors so that \((x, y)\) becomes \((-x, y)\), validating the nature of linear transformations. They offer a framework for systematically applying geometric operations across vector spaces.

Matrix Multiplication

Matrix multiplication is a pivotal operation in linear algebra, enabling the transformation of vectors using matrices. It involves the dot product of rows from the first matrix and columns from the second matrix.
This process is particularly useful in applying transformations such as reflections:

• To multiply a 2x2 matrix by a vector, align the rows of the matrix with the column vector leading to a new vector outcome.
• In reflecting a point around the y-axis, the matrix \(\begin{bmatrix}-1 & 0 \ 0 & 1 \end{bmatrix}\) changes an input vector \((x, y)\) to \((-x, y)\).

Matrix multiplication ensures that transformations are precise and can be applied uniformly across multiple vectors, solidifying its essential role in transforming spatial information securely.