Assume that T is a linear transformation. Find the standard matrix of T.

​T:

set of real numbers R squaredℝ2right arrow→set of real numbers R squaredℝ2​,

rotates points​ (about the​ origin) through

StartFraction 7 pi Over 4 EndFraction7π4

radians.

To find the standard matrix of the linear transformation \( T \), which rotates points in \( \mathbb{R}^2 \) through an angle \( \frac{7\pi}{4} \) radians about the origin, we need to use the matrix representation for a rotation in \( \mathbb{R}^2 \).

### General Formula for a Rotation Matrix:
The standard matrix for a counterclockwise rotation by an angle \( \theta \) is given by:

\[
\begin{bmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix}
\]

In this case, \( \theta = \frac{7\pi}{4} \) radians.

### Step-by-Step Solution:

1. **Find \( \cos\left(\frac{7\pi}{4}\right) \) and \( \sin\left(\frac{7\pi}{4}\right) \):**

   Since \( \frac{7\pi}{4} \) radians is equivalent to \( -\frac{\pi}{4} \) radians (a clockwise rotation by \( \frac{\pi}{4} \)), we can use the known values for the cosine and sine of \( \frac{\pi}{4} \):

   \[
   \cos\left(\frac{7\pi}{4}\right) = \cos\left(-\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
   \]
   \[
   \sin\left(\frac{7\pi}{4}\right) = \sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}
   \]

2. **Construct the rotation matrix using these values:**

\[
\text{Rotation Matrix } = \begin{bmatrix}
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\
-\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{bmatrix}
\]

### Final Answer:

The standard matrix of the linear transformation \( T \) that rotates points by \( \frac{7\pi}{4} \) radians about the origin is:

\[
\boxed{
\begin{bmatrix}
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\
-\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{bmatrix}
}
\]

Would you like more details or have any questions?

Here are some related questions:

1. How do you derive the formula for the rotation matrix?
2. What is the matrix for a clockwise rotation by \( \frac{\pi}{2} \)?
3. How does a reflection matrix differ from a rotation matrix?
4. What is the inverse of a rotation matrix?
5. How do you apply this matrix to transform a point?

**Tip:** When rotating by angles that are multiples of \( \frac{\pi}{4} \), you can simplify calculations by using known sine and cosine values from unit circle symmetries.

Assume that T is a linear transformation. Find the standard matrix of T. T: set of real numbers R² → set of real numbers R², rotates points (about the origin) through 7π/4 radians.

This page provides a step-by-step solution to finding the standard matrix for a linear transformation in ℝ² that rotates points by 7π/4 radians about the origin. Understand the rotation matrix formula and how to apply it. Suitable for undergraduate math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation