Final answer:
The transformation is a linear transformation that could involve rotation and/or scaling. Its matrix representation is a 2x2 matrix with entries derived from the given equations, and the determinant of this matrix is 1, indicating that the transformation is area-preserving.
Step-by-step explanation:
The student has presented a transformation represented by t: x = 5665/3365u − 3365/5665v, y = 3365/5665u + 5665/3365v.
- To identify the transformation type, we can observe that the transformation involves combinations of u and v for both x and y without any trigonometric, exponential or logarithmic functions. This suggests that it's likely a linear transformation, possibly involving rotation and/or scaling.
- The effect on coordinates will depend on the specific values of u and v, but generally, this transformation will map the point (u, v) in the uv-plane to a new point (x, y) in the xy-plane, possibly rotating and scaling the figure represented by a set of points.
- The matrix representation of this transformation is found by rewriting the given equations in matrix form. It can be represented as a 2x2 matrix with the top row being [5665/3365, -3365/5665] and the bottom row being [3365/5665, 5665/3365].
- To calculate the determinant of the transformation matrix, we use the formula ad - bc, where a, b, c, and d are the entries of the 2x2 matrix. For our matrix, the determinant would be (5665/3365)*(5665/3365) - (-3365/5665)*(3365/5665), which simplifies to 1, indicating the transformation is area-preserving.