9, DUE Friday, March 24th, 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 justi\u001ccation 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 \u0010basically getting it right." Cite every theorem or fact from the book that you are using. (\u0010By Theorem 1.10. . . ", etc.) 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 questions from the book: Section 5.4: Section 5.5: 28, 30, 32 8, 16, 20, 24, 26 Part B (25 points) Problem 1. Let A be an n d matrix. (im A) = ker AT to prove that rank(A) = rank(AT ). T that rank(A) = rank(A A). T T or disprove: rank(A A) = rank(AA ). [Hint: parts (a) and (b) are (a) Use the formula (b) Prove (c) Prove Problem 2. Consider the map that associates to each pair real number z = x + iy , of complex numbers the zw + zw , 2 hz, wi = where for any complex number (z, w) useful.] the notation z denotes the complex conjugate z = x iy . (a) Show that this de\u001cnes an inner product on the 2-dimensional vector space 1 C. (b) Find a basis of C that is orthonormal with respect to this inner product. be the inner product on C de\u001cned above, and let be the usual dot 2 2 product on R . Prove that (C, h, i) and (R , ) are 2 . This means that there is an isomorphism T : R C such that for all ~x, ~y R2 , (c) Let h, i isomorphic as inner product spaces ~x ~y = hT (~x), T (~y )i. Problem 3. Let A be an nn diagonal matrix. (a) Show that if the diagonal entries of A are all positive, then the function h~x, ~y i = ~xT A ~y de\u001cnes an inner product on Rn . (b) Show by counterexample that if some diagonal entry of A is non-positive, then T h~x, ~y i = ~x A ~y is not an inner product on R n . Problem 4. In this problem we will consider the following theorem: Theorem: Let V be a \u001cnite dimensional inner product space with basis B. Then there exists a symmetric matrix B such that for all vectors ~x, ~y V , h~x, ~y i = [~x]TB B [~y ]B . (a) Find the matrix B guaranteed to exist by the Theorem for the dot product on Rn with the standard basis. B = (~v1 , . . . , ~vn ) be a matrix whose (i, j)-entry (b) Let basis of the inner product space is h~vi , ~vj i. Show that B where ~x and ~y are both elements of the standard B B be the nn B as in (a) n basis E of R . in the special case in the theorem is unique. (e) Prove the theorem in the case Problem 5. and let is symmetric. (c) Verify that the conclusion of the Theorem holds with (d) Show that the matrix V, n = 3. V be a \u001cnite dimensional inner product space with A = (w ~ 1, . . . , w ~ n ) is another basis of V , and let A be the corresponding matrix whose (i, j)-entry is hw ~ i, w ~ j i. Show that B = S T AS , where S = SBA is the change of coordinates matrix from B to A. basis Just as in Problem 4, let B = (~v1 , . . . , ~vn ). Suppose that