(a) [4 mks) Let Re: R" + R" be a rotation matrix with angle 0 € Sn-1 fixed. Define v:R" + R by v(x) = u(Rox). Show that

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A 4 Mks Let Re R R Be A Rotation Matrix With Angle 0 Sn 1 Fixed Define V R R By V X U Rox Show That 1 (115.3 KiB) Viewed 33 times

(a) [4 mks) Let Re: R" + R" be a rotation matrix with angle 0 € Sn-1 fixed. Define v:R" + R by v(x) = u(Rox). Show that v is a solution to the Laplace equation given that u is a solution to Laplace's equation. You are allowed to state and use standard properties about rotation matrices without proof (finding such standard properties is an exercise of research level Googling skills that mathematicians commonly employ). (b) [4 mks) Write down an example of each of the following properties of solutions to the Laplace equation: (i) invariance under linear combinations, (ii) invariance under rotation in the domain, (iii) invariance under translation in the image, (iv) invariance under translation in the domain, (v) invariance under scaling in the image, and (vi) invariance under scaling in the domain.