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1) if v is contained in V and define the mapping f(x)=<x,v>v. Show f is a linear transformation and describe its range and kernel.

2) if v1 is a subspace of V show that V1 and direct sum of V1 (orthogonal) = V.

I tried to prove these as they were stated in a website. But failed. PLease kindly assist me on this matter.

Thank.