Question

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? ( Please compute both the matrix of the transformation and the change of coordinates matrix!)

(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.

Answer 1