27.2. Examples of Inner Product Spaces

Try these first

1. 1.

   What is the inner product of the vectors (2, 3) and (1, 4)?

   Hint

   Use the formula for the dot product.

2. 2.

   Explain what the complex conjugate is in the context of complex inner products.

   Hint

   Consider what happens to complex numbers when multiplied by their conjugates.

3. 3.

   What is the inner product of the vectors (2, 3) and (1, 4)?

   Hint

   Use the dot product formula.

4. 4.

   True or False: The inner product in complex spaces does not require taking the conjugate of the second vector.

   • True
   • False

   Hint

   Remember how complex numbers behave under multiplication.

5. 5.

   Given two complex vectors u = (1+i, 2-3i) and v = (2, 3+i), find their inner product and interpret the result.

   Hint

   Remember to take the conjugate of vector v.

6. 6.

   Explore the intersection of inner product spaces with an example of orthogonal functions over the interval [0, π]. Show they meet the criteria for having an inner product of zero.

   Hint

   Evaluate this integral to confirm the solution.