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# homework - Homework List for Math 571 (Summer 2011) Kiam...

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Homework List for Math 571 (Summer 2011) Kiam Heong Kwa July 16, 2011 Remark 0. Unless otherwise stated, the exercises are taken from Steven J. Leon, Lin- ear Algebra with Applications 8/e. Remark 1. This list will be updated throughout the quarter. Remark 2. An exercise with an asterisk (*) is relatively more difﬁcult. Remark 3. An exercise in a box is an application problem. Note that the student is responsible for making sure that he/she has access to MAT- LAB during the make-up of any quiz or exam. 5.4 2, 3, 7(c), 9, 10, 11, 17, 18, 20, 21, 22, 24, 26, 28, 29, 30, 32, 33 Exercise 9: Both m and n are assumed to be integers. Exercise 17: Use the Pythagorean law. Also, note that k - y k = k y k . Exercise 18: Adapt the proof of the Pythagorean law. Exercise 20: The nonsingularity of the matrix A is to ensure that k x k A = k A x k 2 = 0 if and only if x = 0 . Exercise 22: First show that k x k 2 2 ≤ k x k 2 1 . Does the result holds for R n , where n is an integer 3 ? Yes. But how? Exercise 26: Simply expand k u + v k 2 = h u + v , u + v i , etc. Exercise 29: For part (b), if x = ( x 1 ,x 2 , ··· ,x n ) , then k x k 2 = n i =1 x 2 i n i =1 max 1 j n x 2 j = n max 1 j n x 2 j = n (max 1 j n | x j | ) 2 . Exercise 30: For part (b), it is the square with vertices ( ± 1 , 0) and (0 , ± 1) . For part (c), it is the square with vertices (1 , ± 1) and ( - 1 , ± 1) . 5.3 1(a), 1(c), 3(a), 4(a), 5, 9, 10, 11, 12, 14 1

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Exercise 10: For part (a), recall that N ( A T ) = R ( A ) by the fundamental subspaces theorem in section 5.2. Also, recall that the system A x = b is consistent if and only if b R ( A ) , the column space of A . For part (b), the system A x = b have inﬁnitely many least squares solutions. To see this, note that the normal system A T A x = A T b = 0 is a homogeneous system and is thus consistent because it has the trivial solution x = 0 . It has inﬁnitely many solutions because rank( A T A ) = rank( A ) = 3 < 5 , while A T A is 5 × 5 . Exercise 11: Recall that an m × m matrix M is called symmetric if M T = M . Exercise 12: First convince yourself that r N ( A T ) = R ( A ) . Then recall that b x is a least squares solution of A x = b if and only if b - A b x R ( A ) . Exercise 14: Refer to application 3 on pp. 229-230 of the text. 5.2 1(c), 1(d), 2, 4, 5, 6*, 7, 9, 11*, 13*, 14, 15, 16* Exercises 2, 4: Adapt the solutions to exercise 3 in the notes. Exercise 5: Use the fundamental subspaces theorem (i.e., theorem 5.2.1) and theorem 5.2.3. Any one dimensional subspace of R 3 is a line passing through the origin, while any two dimensional subsapce of R 3 is a plane passing through the origin. Exercise 6: Let
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## This note was uploaded on 11/11/2011 for the course MATH 571 taught by Professor Kim during the Summer '08 term at Ohio State.

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homework - Homework List for Math 571 (Summer 2011) Kiam...

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