problems-1-223-F07 - M ATH 223 P ROBLEM S ET 1 D UE : 7 S...

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Unformatted text preview: M ATH 223 P ROBLEM S ET 1 D UE : 7 S EPT 4 S EPTEMBER 2007 It was my mistake about the due date on this. If you like, you may hand it in until Friday, 7 September, at my office in Malott with no penalty. Hereafter, problem sets will be due on Tuesdays in lecture. When you hand in this problem set, please indicate on the top of the front page how much time it took you to complete. Reading. 1.11.3. Problems from the book: 1.1.4, 1.1.6 (d) & (g). 1.2.2, 1.2.6. 1.2.8, 1.2.11, 1.2.12, 1.2.22, 1.2.21, 1.2.23. 1.3.2, 1.3.5, 1.3.7, 1.3.12. Additional problems: 1. Suppose that A = a b is a matrix with integer entries. What condition(s) can c d you put on the entries a, b, c, and d to ensure that the inverse A-1 also has integer entries? Is your condition necessary as well as sufficient? a 2. Let - = a be a unit vector in R2. That is, a2 + b2 = 1. b a. Show that the transformation T- : R2 R2 defined by a - ) = - - 2(- - )- T- ( v v a v a a is linear. a v b. What is T- (- )? If - is orthogonal to - , what is T- (- )? Can you describe v a a a T- in general? a c. What is the matrix of T- (in terms of a and b)? a 3. True or False: Determine whether each of the statements is true or false. Please justify your response: explain why it is true, or give an explicit example where the statement fails. 1 a b (a) The matrix A = 0 1 c is invertible. 0 0 1 (b) If A s a square matrix, then AT A = AAT . (c) If A is a square matrix, then AT A is symmetric. ...
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