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Quiz: Inner product spaces

Page history last edited by RH 11 years, 9 months ago

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

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

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

 

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