Introduction to Linear algebra-Strang-Solutions-Manual_ver13

D one solution for all b when a is square and

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online