Final answer:
The correct option is A). A linear transformation mapping the unit circle to an ellipse can be represented by T(x, y) = (ax, by). The transformation scales points on the unit circle by the semi-major axis a and semi-minor axis b of the ellipse, ensuring a one-to-one correspondence between the two shapes.
Step-by-step explanation:
The student has asked for a linear transformation that establishes a one-to-one correspondence between a unit circle and an ellipse. Let us define a unit circle as the set of all points (x, y) such that x² + y² = 1. For an ellipse with semi-major axis a and semi-minor axis b, its equation is (x/a)² + (y/b)² = 1. A linear transformation that maps points from the unit circle to the ellipse can be given by T(x, y) = (ax, by), which effectively scales each x coordinate by a and each y coordinate by b, transforming the unit circle into the given ellipse.
It's important to clarify an incorrect definition: an ellipse is not an open curve as previously stated. An ellipse is a closed curve such that for any point on the curve, the sum of the distances to the two foci is constant. This property is central to constructing an ellipse and understanding its geometry.
An ellipse can be drawn using two pins (representing the foci) and a string, where the length of the string is constant, thereby satisfying the definition. Consequently, a linear transformation that guarantees a one-to-one correspondence will have an inverse transformation, meaning each point on the unit circle maps to exactly one point on the ellipse and vice versa.