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Orthogonality, Orthonormal Basis and Orthogonal Complement

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Please help me with 2 attached problems.

In problem 6 I can imagine the space and guess which vectors are orthogonal to (1,-1,0) like (0,0,1) for example. However, I am not sure of my approach when I do it by hand.

I do not know how to approach problem 12. I am doing something completely wrong and especially need help with this problem.

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Orthogonality, Orthonormal Basis and Orthogonal Complement are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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Please see the attached file.

Note that the vector (0,0,1) is not orthogonal to (1,1,1) - at least not under the standard inner product.

You might want to try it the following way:

Write a general vector ...

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