This quiz is designed to test your knowledge of inner product spaces and related concepts such as inner products, length, orthogonality, and orthonormal bases.

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(Key: correct, incorrect, partially correct.)

1. Let be an inner product space, and let and be two vectors in such that and . What exactly can we say about ?
   1. .
      • INCORRECT. This is true if and are orthogonal (by Pythagoras's theorem), but is not true otherwise.
   2. is less than or equal to 5.
   3. is less than or equal to 7.
      • PARTIALLY. It is true that is less than or equal to 7 (by the triangle inequality), but this is not the only thing one can say about .
   4. is between 0 and 7 inclusive.
      • PARTIALLY. It is true that is less than or equal to 7 (by the triangle inequality), and must be at least 0 (by positivity), but this is not the only thing one can say about .
   5. is equal to 1 or 7.
      • INCORRECT. It is also possible for to take values between 1 and 7. Remember that and are _vectors_, not _numbers_; saying that does not mean that is equal to +3 or -3, and similarly for .
   6. is between 1 and 7 inclusive.
      • CORRECT.
   7. is equal to 7.
2. Let be a complex inner product space, and let and be two vectors in such that and . What exactly can we say about ?
   1. .
      • INCORRECT. This is true if and are orthogonal, but is not true otherwise.
   2. is equal to 12.
   3. is equal to +12 or -12.
      • INCORRECT. and are vectors, not scalars: saying that does not mean that is equal to +3 or -3, and similarly for .
   4. is between -12 and 12 inclusive.
      • PARTIALLY. This is true for real inner product spaces, but for complex inner product spaces can be complex.
   5. is equal to +12, -12, +12i, or -12i.
   6. can be any complex number of magnitude 12 or less.
      • CORRECT.

Score:  

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