Problem 21 - College Algebra with Corequisite Support

Question

For the following exercises, find a new representation of the given equation after rotating through the given angle. \(-2 x^{2}+8 x y+1=0, \theta=45^{\circ}\)

Step-by-Step Solution

Answer

The rotated equation is \(-3x^2 + 4xy + 3y^2 + 1 = 0\).

Step 1: Convert rotation angle to radians

First, convert the rotation angle from degrees to radians because angles in linear algebra are typically expressed in radians. Recall that to convert degrees to radians, you multiply by \( \frac{\pi}{180} \). Thus, \( \theta = 45^{\circ} \) converts to \( \theta = \frac{\pi}{4} \) radians.

Step 2: Rotation Matrix

Write down the 2D rotation matrix corresponding to \( \theta = \frac{\pi}{4} \). This matrix is given by:\[R = \begin{bmatrix}\cos \theta & -\sin \theta \\sin \theta & \cos \theta \end{bmatrix} = \begin{bmatrix}\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\end{bmatrix}\]

Step 3: Convert original equation for rotation

The original equation is a quadratic form \(-2x^2 + 8xy + 1 = 0\). This can be represented in matrix notation as:\[\begin{bmatrix} x & y \end{bmatrix}\begin{bmatrix} -2 & 4 \ 4 & 0 \end{bmatrix}\begin{bmatrix} x \ y \end{bmatrix} + 1 = 0\]where the symmetric matrix represents the coefficients of the quadratic form.

Step 4: Transform matrix with rotation

To find the new coefficients after rotation, compute the similarity transformation using the rotation matrix:\[A' = R^T \cdot A \cdot R\]where:- \( R^T \) is the transpose of the rotation matrix \( R \).- \( A \) is the original coefficient matrix \( \begin{bmatrix} -2 & 4 \ 4 & 0 \end{bmatrix} \).- \( R \) is the rotation matrix computed earlier.

Step 5: Calculate new coefficients

Perform the matrix multiplication step by step. First, compute the transpose of the rotation matrix \( R^T \), then calculate \( R^T \cdot A \) and finally \((R^T \cdot A) \cdot R\). The resulting matrix \( A' \) will give the coefficients for the equation's new orientation. Solving these gives:\[A' = \begin{bmatrix} -3 & 2 \2 & 3 \end{bmatrix}\]

Step 6: Write the rotated equation

Substitute the new coefficients back into the equation form:\[-3x^2 + 4xy + 3y^2 + 1 = 0\]This represents the original equation after the rotation by 45 degrees.

Key Concepts

Rotation Matrix

In the realm of matrix algebra, a rotation matrix serves as the cornerstone for rotating figures in the Cartesian plane. A 2D rotation matrix for an angle \( \theta \) is given by:

• \( R = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \)
• This matrix helps in rotating coordinates without changing their shape or orientation relative to each other.

To rotate a point \((x, y)\) through an angle \(\theta\), you multiply the rotation matrix by the coordinates:

• \( \begin{bmatrix} x' \ y' \end{bmatrix} = R \cdot \begin{bmatrix} x \ y \end{bmatrix} \)

This results in new coordinates \((x', y')\) representing the position after rotation. In this exercise, the rotation matrix for \(45^\circ\), or \(\pi/4\) radians, is used for transforming the given quadratic form.

Quadratic Forms

A quadratic form is an algebraic expression often written as \( ax^2 + bxy + cy^2 = 0 \). It can also be represented in matrix notation for simplification and manipulation:

• The expression is transformed to \( \mathbf{x}^T A \mathbf{x} + d = 0 \), where \( \mathbf{x} = \begin{bmatrix} x \ y \end{bmatrix} \).
• The matrix \( A \) contains the coefficients of the quadratic terms.

In the given problem, the quadratic form \(-2x^2 + 8xy + 1 = 0\) is represented by the symmetric matrix:

• \( A = \begin{bmatrix} -2 & 4 \ 4 & 0 \end{bmatrix} \).

These forms are essential in various fields for closely studying conic sections and other geometric transformations.

Radians Conversion

Radians are vital for expressing angles in most mathematical functions, especially in linear algebra and trigonometry. To convert degrees to radians, you use the conversion factor:

• \( \text{Radians} = \frac{\pi}{180} \times \text{Degrees} \)

For 45 degrees, this conversion is straightforward:

• \( 45^\circ = \frac{\pi}{4} \) radians.

This conversion is necessary because any trigonometric calculations in functions, such as those used in rotation matrices, require angle units to be in radians.

Similarity Transformation

A similarity transformation involves the operation that re-orients a matrix via a specific transformation matrix, without altering the essence of the function it represents. Here, we use this property to rotate the quadratic form matrix:

• The transformation \( A' = R^T A R \) involves the transpose of the rotation matrix \( R \) and the original coefficient matrix \( A \).
• This yields a new orientation matrix \( A' \) with updated coefficients that align with the rotated axes.

In this exercise, the matrix transformation results in a new equation \(-3x^2 + 4xy + 3y^2 + 1 = 0\), showcasing the transformed quadratic form under rotation. This technique is crucial in solving eigenvalue problems and understanding dynamical systems.