(3) Consider the linear map ? : R2 → R2 which sends (1, 0) ↦→ (−3,...

(3) Consider the linear map ? : R2 → R2 which sends (1, 0) ↦→ (−3, 5) and (0, 1) ↦→ (4, −1).

(a) What is the matrix of the transformation? What is the change of coordinates matrix? Do they agree? How come?

(b) Where does this transformation send the area between the vector (4, 2) and the x-axis? Explain algebraically and draw a picture.

(c) What is the image of the lower half plane under ?? Explain algebraically and draw a picture.

(d) What is the pre-image of the upper half plane under ?? Explain algebraically and draw a picture.

(e) What is the image of the unit circle under ?? Explain algebraically and draw a picture.

(f) Deduce whether or not linear transformations preserve angles/areas/curves/shapes, etc.

Expert Solution

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Colby Messinger answered 2 years ago

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