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Inner Product Space, Linear Operators, Commutativity and Standard Inner Product

This is for #2 & 4 section 8.3 in Hoffman and Kunze's Linear Algebra book. Any help and detailed explanations will be greatly appreciated!
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Inner Product Space, Linear Operators, Commutativity and Standard Inner Product are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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