Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May., 2019
Instructor: G. K. Ananthasuresh , Room 106, ME Building, suresh at mecheng.iisc.ernet.in
Lectures: Tu, Th: 08:30 AM - 10:00 AM; Venue: ME MMCR

Homework #3
Assigned: Jan. 29th, 2019
Due: Feb. 5th, 2019
Points: 60
Additional points for work that is beyond instructor's expectation!
Look up Homework #1 of 2010 offering of this course and the solution posted there to know what is meant by "beyond instructor's expectation".

1. 20 points
   A vector space has ordered pairs (x₁, x₂) as its elements. Draw the "unit circle" defined as S around the zero vector (0,0) for the four norms given below (for n = 1, 2, 3, and infinity), all in the same figure.
   
   Attempt the case when n is less than 1 too.
2. 20 points
   If x is perpendicular to y in an inner product space X, show that the following is true.
   
   What you see above is the familiar Pythorean theorem. Extend this formula to m mutually orthogonal vectors and show it to be true.
3. 20 points
   Using an inner product defined by <x,y> below, show that all polynomials P_n are orthogonal to one another. Note that x can be taken as P_i and y as P_j where i and j are integers. For convenience, take small values of i and j. A general verification will fetch extra marks.
   For visualization, draw P_{n = 1 to 6}.