Exercises on projection matrices and least squares
Problem 16.1: (4.3 #17. Introduction to Linear Algebra: Strang) Write down three equations for the line b = C + Dt to go through b = 7 at t = −1, b = 7 at t = 1, and b = 21 at t = 2. Find the least squares solution xˆ = (C, D) and draw the closest line.
Problem 16.2: (4.3 #18.) Find the projection p = Axˆ in the previous problem. This gives the three heights of the closest line. Show that the error vector is e = (2, −6, 4). Why is Pe = 0?
Problem 16.3: (4.3 #19.) Suppose the measurements at t = −1,1,2 are the errors 2, -6, 4 in the previous problem. Compute xˆ and the closest line to these new measurements. Explain the answer: b = (2, −6, 4) is perpendicular to so the projection is p = 0.
Problem 16.4: (4.3 #20.) Suppose the measurements at t = −1,1,2 are b = (5, 13, 17). Compute xˆ and the closest line and e. The error is e = 0 because this b is .
Problem 16.5: (4.3 #21.) Which of the four subspaces contains the error vector e? Which contains p? Which contains xˆ? What is the nullspace of A?
Problem 16.6: (4.3 #22.) Find the best line C + Dt to fit b = 4, 2, −1,0,0 at times t = −2, −1, 0, 1, 2.
