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Chapter 3: Problem 24

Find the standard matrix of the given linear transformation from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\). Reflection in the line \(y=x\)

Short Answer

Expert verified
The standard matrix is \( \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \).

Step by step solution

01

Understanding Reflection Rules

To find the standard matrix for reflection across the line \( y = x \), we need to understand that this transformation swaps the x-coordinate and y-coordinate of each point in the plane.
02

Matrix Representation of Reflection

The standard matrix for this type of reflection is \( \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \) because this matrix swaps the coordinates of any vector \( \begin{bmatrix} x \ y \end{bmatrix} \) to \( \begin{bmatrix} y \ x \end{bmatrix} \).
03

Verify the Transformation

Apply the matrix \( \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \) to a vector \( \begin{bmatrix} x \ y \end{bmatrix} \). The resulting vector is \( \begin{bmatrix} 0 \cdot x + 1 \cdot y \ 1 \cdot x + 0 \cdot y \end{bmatrix} = \begin{bmatrix} y \ x \end{bmatrix} \), confirming the reflection across the line \( y = x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflection Matrix
A reflection matrix is a specific type of matrix used in linear algebra to perform a reflection transformation. In the plane, a reflection matrix creates a mirror image of points across a given line. This type of transformation is an example of a linear transformation that maintains the distance and angles of shapes, but flips them over a line of symmetry. The result of applying a reflection matrix is a change in the orientation of points, creating a mirror effect.
  • Reflection keeps the size and shape of the figures identical post-transformation.
  • It involves swapping or negating coordinates based on the line of reflection.
  • Useful in computer graphics and geometrical modeling for creating symmetrical designs.
The reflection matrix simplifies the process of calculating images across lines, making it a vital tool in geometry and physics. Understanding how to construct and use these matrices is essential for solving complex reflection problems easily.
Standard Matrix
The standard matrix of a linear transformation is a specific matrix that represents the transformation in a consistent form. This matrix acts as a blueprint for how a given transformation changes each vector that it is applied to. For reflections specifically, the standard matrix translates a verbal rule (like swapping x and y) into a mathematical operation.
  • It adheres to a prescribed format, making calculations systematic and reliable.
  • Determined by applying the transformation to basis vectors.
  • It allows for quick verification and application across various problems.
For reflection across the line \(y = x\), the standard matrix simplifies to \( \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \). Applying this matrix to any vector rearranges its coordinates, providing a simple solution to the reflection operation.
Reflection in the line y=x
Reflecting a point across the line \(y = x\) involves flipping its coordinates. Points that were originally on this line remain unchanged, while others transform to their mirrored position. This is a fundamental concept in mathematics, often used to test understanding of coordinate geometry.
  • The line \(y = x\) is a diagonal line running through the origin with a 45-degree angle.
  • By swapping x and y coordinates, points are moved perpendicularly across this line.
  • Applicable in solving symmetrical problems, especially in algebra and calculus.
By applying the reflection matrix \( \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \), any vector \( \begin{bmatrix} x \ y \end{bmatrix} \) is transformed to \( \begin{bmatrix} y \ x \end{bmatrix} \). This method is consistent and straightforward, offering a reliable approach to geometric transformations.

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Most popular questions from this chapter

Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). $$\left[\begin{array}{ll}0 & 1 \\\1 & 1\end{array}\right] \text { over } \mathbb{Z}_{2}$$

Find the standard matrix of the composite transformation from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\). Clockwise rotation through \(45^{\circ},\) followed by projection onto the \(y\) -axis, followed by clockwise rotation through \(45^{\circ}\).

Let \(\ell\) be a line through the origin in \(\mathbb{R}^{2}, P_{\ell}\) the linear transformation that projects a vector onto \(\ell\), and \(F_{\ell}\) the transformation that reflects a vector in \(\ell\) (a) Draw diagrams to show that \(F_{\ell}\) is linear. (b) Figure 3.14 suggests a way to find the matrix of \(F_{c}\) using the fact that the diagonals of a parallelogram bisect each other. Prove that \(F_{\ell}(\mathbf{x})=2 P_{\ell}(\mathbf{x})-\mathbf{x}\) and use this result to show that the standard matrix of \(F_{\ell}\) is \\[ \frac{1}{d_{1}^{2}+d_{2}^{2}}\left[\begin{array}{cc} d_{1}^{2}-d_{2}^{2} & 2 d_{1} d_{2} \\ 2 d_{1} d_{2} & -d_{1}^{2}+d_{2}^{2} \end{array}\right] \\] (where the direction vector of \(\ell\) is \(\mathrm{d}=\left[\begin{array}{l}d_{1} \\ d_{2}\end{array}\right]\) ). (c) If the angle between \(\ell\) and the positive \(x\) -axis is \(\theta\) show that the matrix of \(F_{\ell}\) is \\[ \left[\begin{array}{rr} \cos 2 \theta & \sin 2 \theta \\ \sin 2 \theta & -\cos 2 \theta \end{array}\right] \\]

Write the given permutation matrix as a product of elementary (row interchange) matrices. $$\left[\begin{array}{ccccc} 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \end{array}\right]$$

Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). $$\left[\begin{array}{rr}-2 & 4 \\\3 & -1\end{array}\right]$$
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