(Solution) - Let and let E y Ax x -(2025 Original AI-Free Solution)

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Academic Level: Undergrad. (yrs 3-4)

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Let

and let E = {y = Ax | ||x|| = 1} be the image of the unit Euclidean sphere under the linear map induced by A.
(a) Explain why E is an ellipsoid and write down its equation.
(b) What are its principal axes and their lengths- the semi-axes of the ellipsoid?
(c) What is the volume of the solid ellipsoidal domain enclosed by E?