27.3. Norm Induced by Inner Product

Try these first

1. 1.

   What is the formula for the norm of a vector in an inner product space?

   Hint

   Recall how the inner product is used to determine the norm.

2. 2.

   Is the norm ever negative? Why or why not?

   Hint

   Think about the properties of lengths or distances.

3. 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

   Hint

   Think about how we calculate the distance or length.

4. 4.

   True or False: The norm of a vector can be negative.

   • True
   • False

   Hint

   What do you know about lengths?

5. 5.

   Given vector a = (3, 4) and b = (1, 2), calculate the distance between these two vectors using their norms.

   Hint

   Remember that distance can be found using the norm of the difference between the vectors.

6. 6.

   Prove that the norms behave correctly under scalar multiplication by showing how ∥2v∥ relates to ∥v∥.

   Hint

   Use the definition of the norm with the inner product to show this relationship.