As an expert in the field of linear algebra and matrix transformations, I'm delighted to provide you with a comprehensive understanding of the concept of a reflection matrix and how it is used to reflect figures across lines of symmetry.

A reflection matrix is a type of linear transformation matrix that reflects a vector across a line, a point, or a plane in a given vector space. This transformation is a fundamental concept in geometry and has wide applications in various fields such as computer graphics, physics, and engineering.

### Definition and Properties

A reflection matrix, denoted as \( R \), is a square matrix that is orthogonal, meaning that its inverse is equal to its transpose, \( R^{-1} = R^T \). The determinant of a reflection matrix is \( \pm 1 \), which indicates that it represents a rigid transformation (no scaling or rotation other than the reflection itself).

### Construction of a Reflection Matrix

To construct a reflection matrix, you need to identify the line or plane of reflection. For a reflection across a line, this line must be specified by a normal vector \( \mathbf{n} \) that is perpendicular to the line of reflection. The reflection matrix can then be constructed using the following formula:

\[ R = I - 2\frac{\mathbf{n}\mathbf{n}^T}{\mathbf{n}^T\mathbf{n}} \]

Here, \( I \) is the identity matrix of the same dimension as the space, and \( \mathbf{n}\mathbf{n}^T \) is the outer product of the normal vector with itself.

### Example

Let's consider a two-dimensional space and a reflection across the line \( y = x \). The normal vector to this line is \( \mathbf{n} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \). The reflection matrix \( R \) for this line would be:

\[ R = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - 2\frac{\begin{bmatrix} 1 \\ 1 \end{bmatrix}\begin{bmatrix} 1 & 1 \end{bmatrix}}{1^2 + 1^2} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - 2\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} = \begin{bmatrix} -1 & -1 \\ -1 & -1 \end{bmatrix} \]

### Application in Reflecting a Vector

To reflect a vector \( \mathbf{v} \) across a line or plane, you simply multiply the reflection matrix \( R \) by the vector \( \mathbf{v} \):

\[ \mathbf{v}' = R\mathbf{v} \]

The resulting vector \( \mathbf{v}' \) is the reflection of \( \mathbf{v} \) across the specified line or plane.

### Reflection Across a Point

If the reflection is across a point, the reflection matrix is slightly different. It involves negating the coordinates of the vector relative to the point of reflection.

### Conclusion

Reflection matrices are a powerful tool for understanding and performing geometric transformations. They are particularly useful in computer graphics for mirroring images and in physics for modeling symmetrical systems. Understanding how to construct and apply these matrices is crucial for anyone working with transformations in a mathematical or computational context.

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Transformation of Graphs Using Matrices - Reflection. A reflection is a transformation representing a flip of a figure. Figures may be reflected in a point, a line, or a plane. ... Use the following rule to find the reflected image across a line of symmetry using a reflection matrix.read more >>