In the given figure, the vectors u and v are related as \( Au = v \) by a transformation matrix A. The correct choice of matrix A is
% Vector details
u = (4,3), v = (5,0)
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A transformation that preserves the length of a vector is a rotation. The matrix for a counter-clockwise rotation by angle \( \theta \) is \( \begin{pmatrix} \cos\theta & -\sin\theta \sin\theta & \cos\theta \end{pmatrix} \).
Step 1: Analyze vectors \( u \) and \( v \). Given \( u = \begin{pmatrix} 4 \\ 3 \end{pmatrix} \) and \( v = \begin{pmatrix} 5 \\ 0 \end{pmatrix} \), calculate their magnitudes: \( ||u|| = \sqrt{4^2 + 3^2} = 5 \). \( ||v|| = \sqrt{5^2 + 0^2} = 5 \). Since \( ||u|| = ||v|| \), the transformation from \( u \) to \( v \) is a rotation.
Step 2: Find the rotation angle. Vector \( u \) forms an angle \( \theta \) with the x-axis, where \( \cos\theta = \frac{4}{5} \) and \( \sin\theta = \frac{3}{5} \). Vector \( v \) lies along the x-axis (angle 0). The rotation angle from \( u \) to \( v \) is \( -\theta \).
Step 3: Define the rotation matrix. A rotation matrix for angle \( \alpha \) is \( R(\alpha) = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix} \). Here, \( \alpha = -\theta \), so \( \cos(-\theta) = \frac{4}{5} \) and \( \sin(-\theta) = -\frac{3}{5} \). Therefore, the transformation matrix \( A \) is: \[ A = R(-\theta) = \begin{pmatrix} \cos(-\theta) & -\sin(-\theta) \\ \sin(-\theta) & \cos(-\theta) \end{pmatrix} = \begin{pmatrix} \frac{4}{5} & \frac{3}{5} \\ -\frac{3}{5} & \frac{4}{5} \end{pmatrix} \] Step 4: Validate the result. Compute \( Au \): \[ Au = \begin{pmatrix} \frac{4}{5} & \frac{3}{5} \\ -\frac{3}{5} & \frac{4}{5} \end{pmatrix} \begin{pmatrix} 4 \\ 3 \end{pmatrix} = \begin{pmatrix} 5 \\ 0 \end{pmatrix} = v \] The result confirms that the matrix \( A \) correctly transforms \( u \) to \( v \).
