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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online