Introduction to Linear algebra-Strang-Solutions-Manual_ver13

D one solution for all b when a is square and

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Unformatted text preview: ) Any solution can be xp  p 1 6 33 x 2 is shorter (length 2) than D . Then (c) (length 2) 6 1 33 y 0 (d) The only “homogeneous” solution in the nullspace is x n D 0 when A is invertible. Solutions to Exercises 35 14 If column 5 has no pivot, x5 is a free variable. The zero vector is not the only solution 15 16 17 18 19 20 21 22 23 24 25 26 to Ax D 0. If this system Ax D b has a solution, it has infinitely many solutions. If row 3 of U has no pivot, that is a zero row. U x D c is only solvable provided c 3 D 0. Ax D b might not be solvable, because U may have other zero rows needing more ci D 0. The largest rank is 3. Then there is a pivot in every row. The solution always exists. The column space is R3 . An example is A D Œ I F  for any 3 by 2 matrix F . The largest rank of a 6 by 4 matrix is 4. Then there is a pivot in every column. The solution is unique. The nullspace contains only the zero vector. An example is A D R D Œ I F  for any 4 by 2 matrix F . Rank D 2; rank D 3 unless q D 2 (then rank D 2). Transpose has the same rank! Both matrices A have rank 2. Always AT A and AAT have the same rank as A. " #" #    100 10 1 0 10 3 410 02 2 3. A D LU D I A D LU 2 1 0 21 0 301 031 0 0 11 5 "# "# "# "# "# "# "# x 4 1 1 x 4 1 1 Cz 0 0 . The second (a) y D 0 C y (b) y D 0 C z z 0 0 1 z 0 1 equation in part (b) removed one special solution. If Ax 1 D b and also Ax 2 D b then we can add x 1 x 2 to any solution of Ax D B : the solution x is not unique. But there will be no solution to Ax D B if B is not in the column space. For A; q D 3 gives rank 1, every other q gives rank 2. For B; q D 6 gives rank 1, every other q gives rank 2. These matrices cannot have rank 3.      11 b1 x1 1 Œx  D has 0 or 1 solutions, depending on b (b) D (a) 1 b2 x2 Œ b  has infinitely many solutions for every b (c) There are 0 or 1 solutions when A has rank r < m and r < n: the simplest example is a zero matrix. (d) one solution for all b when A is square and invertible (like A D I ). (a) r < m, always r  n (b) r D m, r < n (c) r < m; r D n (d) r D m D n. # " # " # " 10 2 244 244 0 3 6 !RD 0 1 2 and 0 3 6 ! R D I . 005 000 00 0 27 If U has n pivots, then R has n pivots equal to 1. Zeros above and below those pivots make R D I . " #       2 120 1235 1230 1200 1; ! ! ; xn D 28 001 0048 0040 0010 0 Free x2 D 0 gives x p D . 1; 0; 2/ because the pivot columns contain I . " # "# " 1000 0 10 29 Œ R d  D 0 0 1 0 leads to x n D 1 ; Œ R d  D 0 0 0000 0 00 no solution because of the 3rd equation  1 . 2 0 1 0 # 1 2: 5 Solutions to Exercises 36 23 23 # 4 2 4 6 37 6 07 3 ; 4 5; x n D x3 4 5. 30 0 1 2 2 0 " # "#  11 1 0 31 For A D 0 2 , the only solution to Ax D 2 is x D . B cannot exist since 1 03 3 2 equations in 3 unknowns cannot have a unique solution. 2 3 2 32 3 131 1 1 31 1 27 61 2 37 61 1 7 60 32 A D 4 factors into LU D 4 and the rank 5 40 2 4 65 221 0 05 115 1201...
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This note was uploaded on 09/25/2012 for the course PHY 103 taught by Professor Minki during the Spring '12 term at Korea Advanced Institute of Science and Technology.

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