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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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    Solution Preview

    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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