Question

Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is a linear transformation. Let hom (V, W) be the set of linear transfomnations frorn V to W. honn(U,,W) (i-1.2) be the set of linear transformations from U, (z-1.2) to 11 Show that the funetion hom(Ui. W)x hom(U2. W)hom(V, W) defined by (.)- where f(u u2) )+2(u2) is actually a bijection of sets. METHOD OF LINEAR EXTENSION Let V be a finite dimensional vector space, dim V-n and B-(ul. "2-. . . .%) be a basis of V Let W be a vector space Given any function fun, u2 เ«J W. there exists a unique linear transformation VW such that (u) f(u) for each i 1,2..n WHAT IS THE PURPOSE OF THE METHOD OF LINEAR EXTENSION? The whole purpose of this method of linear extension is: defining linear transformations on finite dimensional vector spaces is as simple as defining functions on finite sets. How shall we utilise this? Given the basis B(,2 .. n) of V, select n vectors (u) f(u2) ....(u) fron W (another vector space) and then using the method above there is the unique linear transformation VW defined by: (e) Let V be a fnite dimensional vector space, U be a vector subspace ofV, W be a vector space and UW be a linear transformation 20

Answer 1

(a) Let f: R^2 rightarrow R be a function such that f(x, y) = x + y (x, y) elementof R^2 lambda(x, y) = (lambda x, lambda y) f(lambda(x, y)) = f(lambda x, lambda y) = lambda x + lambda y = lambda(x + y) = lambda f(x, y) x = (x, y) elementof R^2 f(lambda x) = lambda f(x) (b) Given that V is 1-dim vector space and L: V rightarrow W be a linear transformation. Basis {u_1} of V must be transform basis {L(u_1)} of W every vector of V can be express as lambda u_1 L(lambda u_1) = lambda L(u_1) Because L is linear transformat? We can easily determine every transform image of vector space V by using scalar lambda (any) in the field (c) Given that V = U_1 U_2 f_1: U_1 rightarrow W & f_2: U_2 rightarrow W be a linear transformation We have to show that f: V rightarrow W define by