27.11. Best Approximation in Inner Product Spaces

Try these first

1. 1.

   Define best approximation in your own words.

   Hint

   Think about it as finding the nearest point.

2. 2.

   What does the error vector signify?

   Hint

   Look for a relationship between the original and approximated values.

3. 3.

   What is the best way to define best approximation?

   • The farthest vector from the original
   • The closest vector in a subspace
   • Any random vector from the space

   Hint

   Refer back to the definition discussed in class.

4. 4.

   The error vector is known for being orthogonal to what?

   • True
   • False

   Hint

   Remember the definition of the Projection Theorem.

5. 5.

   Given a vector v = (4, 3) and a subspace defined by w1 = (1, 1) and w2 = (1, -1), find the best approximation vector v_b.

   Hint

   Break down the steps using projection formulas.

6. 6.

   In a practical setting, if you have a data set with noise and want to fit your model optimally using least squares, how would you identify the best parameters?

   Hint

   Utilize derivatives to find minima in your functions.