Math 5311 – Problem Set #3

Due Tuesday, 3 Feb 2009

Problem 1

Olver & Shakiban 4.3.3 asked you to find the point on the plane x + 2y − z = 0 that is closest to the point(0, 0, 1)T. O&S solve such problems by finding a basis for the subspace defined by the plane Ax = 0, thenapplying Theorem 4.5 to minimize the distance in that subspace. See the solution set for PS 1 for a workedsolution using that method.

Here’s another method of solution that lets you bypass the need to find a basis for ker(A): minimize the dis-

tance ‖x − p‖2 over x ∈ R3, subject to the constraint that Ax = 0. You should have learned in a calculus course

the method of Lagrange multipliers for solving constrained minimization problems (e.g., Strauss, Bradley, andSmith, Calculus, Ch 11.8).

1. Use the method of Lagrange multipliers to minimize d(x, p)2 = x2 + y2 +(z− 1)2 subject to the constraintx + 2y − z = 0.

We will soon develop a similar approach to solving boundary-value problems by treating the boundary con-ditions as constraints, bypassing the need to find a basis for the subspace H1

0 .

1