Practice Norm Induced By Inner Product (27.3) - Inner Product Spaces

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Exercise
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6 sample questions

Question 1.
What is the formula for the norm of a vector in an inner product space?
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Recall how the inner product is used to determine the norm.
Question 2.
Is the norm ever negative? Why or why not?
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Think about the properties of lengths or distances.
3Question 3.
What is the norm of a vector defined as?
- The inner product of the vector with itself
- The outer product of the vector
- The sum of components of the vector
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Think about how we calculate the distance or length.
4Question 4.
True or False: The norm of a vector can be negative.
- True
- False
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What do you know about lengths?
Question 5.
Given vector a = (3, 4) and b = (1, 2), calculate the distance between these two vectors using their norms.
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Remember that distance can be found using the norm of the difference between the vectors.
Question 6.
Prove that the norms behave correctly under scalar multiplication by showing how ∥2v∥ relates to ∥v∥.
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Use the definition of the norm with the inner product to show this relationship.

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