Math Problem Statement
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.
Solution
To find the standard matrix of the linear transformation , which rotates points in through an angle radians about the origin, we need to use the matrix representation for a rotation in .
General Formula for a Rotation Matrix:
The standard matrix for a counterclockwise rotation by an angle is given by:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Transformation
Rotation in ℝ²
Matrix Representation
Formulas
Rotation Matrix: [cos(θ) -sin(θ); sin(θ) cos(θ)]
Theorems
Rotation Theorem in ℝ²
Suitable Grade Level
Undergraduate Mathematics