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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.)

- Let be an inner product space, and let and be two vectors in such that and . What exactly can we say about ?
- .
- INCORRECT. This is true if and are orthogonal (by Pythagoras's theorem), but is not true otherwise.

- is less than or equal to 5.
- 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 .

- 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 .

- 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 .

- is between 1 and 7 inclusive.
- is equal to 7.

- Let be a complex inner product space, and let and be two vectors in such that and . What exactly can we say about ?
- .
- INCORRECT. This is true if and are orthogonal, but is not true otherwise.

- is equal to 12.
- 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 .

- is between -12 and 12 inclusive.
- PARTIALLY. This is true for real inner product spaces, but for complex inner product spaces can be complex.

- is equal to +12, -12, +12i, or -12i.
- can be any complex number of magnitude 12 or less.

## Score:

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