MATH 221 Written Assignment 3 Due Thursday, February 23, 2017 in class Question numbers refer to the exercises at the end of each section of the textbook (1st edition numbers are given if they are different). You must show your work to get full marks. For some questions, I have given hints, clarifications, or extra instructions. Be sure to follow these! Note: Only a selection of exercises may be graded. 1) ! 3 ! 7 ! 1 Let u = 1 , v = 5 , w = 4 . 5 1 8 ! 1! 3 a. Calculate 2 u + 2 v and comment on what you notice about the result in terms of ! the vector w . ! ! ! b. Based on your observation in a., show that the set {u, v, w} is linearly dependent by writing down a linear dependence relation (a non-trivial linear combination ! ! ! that gives the zero vector) on u, v, w . 3 c. Now let A = 1 5 ! x $ ! # 1 & vector x = # x2 & #" x3 &% 7 1 5 4 . Based on your answer in part b., give a non-zero 1 8 ! ! such that Ax = 0 . 2) Do 2.3 #28. Show all your computations to justify your answer. 3) Do 2.3 #48(b) (#44 in 1st ed.). State clearly if this is True or False and make sure to justify your answer. If it is false, give a specific counterexample to the claim. If it is true, you must explain why it is true in general. 4) (Based on 3.1 #20). 3x2 x 1 2 3 Consider the transformation T from R to R given by T = x1 + 5 x2 . x2 2x1 Show that T is not linear by finding at least one specific numerical counterexample to one of the defining properties of a linear transformation. 5) 1 3 ! ! Let T ( x ) = Ax be the linear transformation defined by the matrix A = 2 1 . 5 1 For the following questions, use theorems and do as few computations as possible (or none) to justify your answers. a. Is T one-to-one? (If not, find two different vectors in the domain that have the same image.) b. Is T onto? (If not, find a vector in the codomain that is not in the range of T.) 6) x x 1 1 Suppose that T : R R is given by T x2 = x2 . This is called a x 0 3 projection transformation because it projects a vector in R3 onto the plane x3 = 0 . 3 3 a. Prove that T satisfies the definition of a linear transformation. ! ! b. Find the matrix A such that T ( x ) = Ax (the standard matrix of T). ! ! c. Describe the set of vectors such that T ( x ) = 0 , both as a set and geometrically in R3. (This set is called the kernel of T.) x 1 x d. The transformation P x2 = 1 is similar to T, but goes from R3 to R2. x x2 3 What is the matrix for the transformation P? 7) ! ! Do 3.1 #60. [Hint: Show that T (u1 ) and T (u2 ) must have a dependence relation.] This says that any linear transformation must take dependent vectors in the domain to dependent vectors in the codomain. Bonus: Find an example of a linear transformation T and two independent vectors ! ! ! ! u, v such that T (u ) and T ( v ) are not independent