University:
Whitman College
Course:
MATH-240 | Linear Algebra
Academic year:

2020
Views:

202

Pages:

2
Author:

rion7nd

Summary: Making Nonlinear Functions into (Locally) Linear Functions We denote a nonlinear function F : IRn → IRm by its coordinate functions,  f1 (x1 , x2 , . . . , xn ) f2 (x1 , x2 , . . . , xn ) .. .   y = F (x) =          fm (x1 , x2 , . . . , xn ) where fi : IRn → IR1 . The Jacobian of F is an m × n array of partial derivatives:    J(F (x)) =    (f1 )x1 (f2 )x1 .. . (f1 )x2 (f2 )x2 . . . (f1 )xn . . . (f2 )xn .. .       (fm )x1 (fm )x2 . . . (fm )xn Note that the ith row of J(F ) is the gradient of fi . From Calculus I, if y = f (x), then the local linearization of f at (a, f (a)) is the equation of the tangent line, y − f (a) = f 0 (a)(x − a) Now, if F : IRn → IRm , the local linearization of F about a vector a is: y − F (a) = J(F (a))(x − a) If we let v = y − F (a) and u = x − a, then this is: v = J(F (a))v or “Ax = b00 Special Cases: • If F : IR1 → IRn (these are “parametric equations”, or “space curves”), then the Jacobian collapses to a single column of regular derivatives, and the tangent line at a is:   0 f1 (a)   0  f2 (a)  y − F (a) =  ..   (t − a)   .  fn0 (a) • If F : IRn → IR1 , then the Jacobian collapses to a single row (to the gradient of F ), and the local linearization is the tangent plane: y − F (a) = [ fx1 (a) fx2 (a) . . . fxn (a) ](x − a) (Note the sizes of things here!) EXERCISES: 1. Find the general Jacobian matrix for each function • F : IR2 → IR3 where: x2 + sin(x)   F (x, y) =  x(y − 2)  y 2 − 3xy  • G : IR3 → IR2 where:  " F (x, y, z) = x2 − y 2 3xyz − 5 # 2. For the first function, F, in the previous problem, find the linearization of F at x = (0, 1). Use this to approximate F (1/2, 1/2), and check your answer by actually computing F (1/2, 1/2). 3. For the second function, G, in Problem 1, find the linearization of G at x = (1, 1, 1). Use this to approximate G((1, 0, 1)) and check your answer by computing G((1, 0, 1)). 4. Is the local linear map for F 1-1? Onto? 5. Is the local linear map for G 1-1? Onto? 6. Let F : IR1 → IR3 by:   cos(t)   F (t) =  sin(t)  t Find an equation for the tangent line to the graph of F at t = π/2. 7. Let F : IR3 → IR1 by: F (x, y, z) = 3xy − y 2 + 4 − yz Find an equation for the tangent “plane” to the graph of F at (1, 1, 1) (the quotes are included because the graph of F is in 4 dimensions). 8. For the given function F : IR2 → IR2 , first solve for (x, y) such that F (x, y) = 0, then linearize F about these points: " F (x, y) = x(3 − x − 2y) y(2 − x − y) # Note: You need to solve these systems simultaneously. For example, for the first coordinate to be zero, we need either x = 0 or x = 3 − 2y. Use the second equation now to solve for the corresponding y’s.

Summary Making Nonlinear Functions into (Locally) Linear Functions

Related Documents

Recommended Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock