1. Consider the transformation F :R2 > R2 dened by F (a), y) = (m2 + 2mg 4, my 1). You'll notice that we've often been using the letter T rather than F, and 93,3; as dependent variables with independent variables um when writing planar transformation. This helps prepare our use of transformations for change of variables later in double integrals, but here we will use notation that highlights the similarities with the case of a function f : R > IR. In fact, rather than using coordinate notation for a point in R2, it's convenient to instead use notation as a column vector (Le. 2 x 1 matrix), which allows us to use matrix operations. Everything that follows can be translated into coordinate notation if needed, however. So Fmaps [ 3 ] H [ f1($,y) ] where f1(:c,y) =$2+2$y4 and f2(a;,y) =my 1, (a) Compute the J aeobian matrix of F: 65\" 95\" JF (x) ' an an a: By (b) A key property of the J acobian matrix is that it providae the best linear approximation of F locally around a given point xo = [ :3 ] in the sense that for the transformation L : R2 -> R2 dened by M") = F(X0) + JleoXX - Xe) approximates F for x nearby x0 F(x) 3:: L(x) better than any other transformation of the form ' :1: (11:1: + bly + 01 H [y] [a2w+b2y+02 (1) 1 . . Compute L(x) around x0 = [ 3 ] and Simplify the expression to be in the form of (1). E _ 1.1 . (c) valuate L(a) where a [ 2.8 ] and compare With F(a). (Use a calculator / phone). ,,,1______.L LLA 4.\".A .. ' ' nMAk'Q \"orclnn AF Npqntnn'g