MATH 217 - LINEAR ALGEBRA HOMEWORK 8, DUE Friday, March 17th, 2017 Hand in Part A and Part B as two separate assignments. Include the following information in the top left corner of every assignment: your full name, instructor's last name and section number, homework number, whether they are Part A problems or Part B problems. A few words about solution writing: Unless you are explicitly told otherwise for a particular problem, you are always expected to show your work and to give justification for your answers. In particular, when asked if a statement is true or false, you will need to explain why it is true or false to receive full credit. Write down your solutions in full, as if you were writing them for another student in the class to read and understand. Don't be sloppy, since your solutions will be judged on precision and completeness and not merely on \"basically getting it right\". Cite every theorem or fact from the book that you are using (e.g. \"By Theorem 1.10 ...\"). If you compute something by observation, say so and make sure that your imaginary fellow student who is reading your proof can also clearly see what you are claiming. Justify each step in writing and leave nothing to the imagination. Part A (15 points) Solve the following problems from the book: Section 5.1: 26, 42. Section 5.2: 12, 22, 32. Section 5.3: 35, 40, 54. Part B (25 points) Problem 1. Recall the Cauchy-Schwartz inequality: for all ~x, ~y Rn , |~x ~y | k~xkk~y k. (a) Prove that the square of the average of a list of numbers is less than or equal to the average of the squares of the numbers. That is, show that for all x1 , . . . , xn R, !2 n n 1X 1X 2 xi xi . n i=1 n i=1 (b) Without using calculus, prove that for all R, cos() + sin() 1 2. MATH 217 - LINEAR ALGEBRA HOMEWORK 8, DUE Friday, March 17th, 2017 2 Problem 2. Let T : Rn Rm be any linear transformation, and let S : ker(T ) im(T ) be the linear transformation from ker(T ) to im(T ) defined by S(~x) = T (~x) for all ~x ker(T ) . Prove that S is an isomorphism. Problem 3. The three points ! 2 2 3, + 3, 1 , Q P 2 2 ! 2 2 1, + 1, 1 , 2 2 \u0010 \u0011 and R 2 3 3, 2 + 3 + 3, 2 determine a plane in R3 , and inside this plane the points are successive vertices of a regular polygon (not necessarily in the given order). How many sides does this polygon have? Be sure to explain how you arrive at your answer. Problem 4. Recall that the complex numbers C and the set R22 of 2 2 matrices both \u0014 \u0015 form a b vector spaces. Let V be the subspace of R22 consisting of all matrices of the form where b a a, b R. Define the transformation T : C V by \u0014 \u0015 a b T (a + bi) = . b a Then T is in fact an isomorphism between C and V (you do not have to prove this). (a) Prove that under the above isomorphism, multiplication of complex numbers corresponds to multiplication of 2 2 matrices. In other words, show that for all , C, T () = T ()T (). (b) Under the isomorphism T , which complex numbers correspond to orthogonal matrices? Which operation on C corresponds to taking the transpose of a 2 2 matrix? (c) For an orthogonal matrix A V , we have AT = A1 . What is the corresponding statement for C? Can you provide a proof using complex numbers? Problem 5. Let T : Rn Rm be a linear transformation. Then there exists a unique linear transformation T : Rm Rn , called an adjoint linear transformation of T , with the following property: T (~x) ~y = ~x T (~y ) for all ~x Rn and ~y Rm . (You may assume without proof that such a transformation T exists and is unique). (a) Prove that for all ~v im(T ) and w ~ ker(T ), ~v w ~ = 0. k n (b) Letting S : R R be another linear transformation, prove that (T S) = S T . (c) Letting A be the standard matrix of T , what is the standard matrix of T ? (d) Find the adjoint linear transformation of a general rotation about a line in R3 , and of a general reflection through a plane in R3