Introduction to Linear algebra-Strang-Solutions-Manual_ver13

C ad d bc will make the forward matrix singular and

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Unformatted text preview:    2 2 00 15 This A is not invertible. AM D I is impossible. A D . The range 1 1 00 contains only matrices AM whose columns are multiples of .1; 3/.     00 01 16 No matrix A gives A D . To professors: Linear transformations on 10 00 matrix space come from 4 by 4 matrices. Those in Problems 13–15 were special. M DA 1 17 For T .M / D M T (a) T 2 D I is True (d) False.   0b a0 18 T .I / D 0 but M D D T .M /; these M ’s fill the range. Every M D 00 cd is in the kernel. Notice that dim (range) C dim (kernel) D 3 C 1 D dim (input space of 2 by 2 M ’s).  19 T .T 1 .M // D M so T (b) True  1 .M / D A 1 MB 1 (c) True . 20 (a) Horizontal lines stay horizontal, vertical lines stay vertical (b) House squashes onto a line (c) Vertical lines stay vertical because T .1; 0/ D .a11 ; 0/.     :7 :7 20 projects the house (since 21 D D doubles the width of the house. A D 01 :3 :3 A2 D A from trace D 1 and   0; 1 The projection is onto the column space of D ). 11 A D line through .:7; :3/. U D will shear the house horizontally: The point 01 at .x; y/ moves over to .x C y; y/. Solutions to Exercises 77   0 with d > 0 leaves the house AH sitting straight up (b) A D 3I d   cos  sin  expands the house by 3 (c) A D rotates the house. sin  cos  a 22 (a) A D 0 23 T .v/ D T .v / D v rotates the house by 180ı around the origin. Then the affine transformation v C .1; 0/ shifts the rotated house one unit to the right. 24 A code to add a chimney will be gratefully received! 10 10 10 10  10 :5 :5 26 compresses vertical distances by 10 to 1. projects onto the 45ı line. 0 :1 :5 :5   p :5 :5 rotates by 45ı clockwise and contracts by a factor of 2 (the columns have :5 :5   p 11 length 1= 2). has determinant 1 so the house is “flipped and sheared.” One 10 way to see this is to factor the matrix as LDLT :      11 10 1 11 D D (shear) (flip left-right) (shear): 10 11 1 01 25 This code needs a correction: add spaces between    27 Also 30 emphasizes that circles are transformed to ellipses (see figure in Section 6.7). 28 A code that adds two eyes and a smile will be included here with public credit given! 29 (a) ad bc D 0 (b) ad b c > 0 (c) jad b c j D 1. If vectors to two corners transform to themselves then by linearity T D I . (Fails if one corner is .0; 0/.) v1 30 The circle v2 transforms to the ellipse by rotating 30ı and stretching the first axis by 2. 31 Linear transformations keep straight lines straight! And two parallel edges of a square (edges differing by a fixed v) go to two parallel edges (edges differing by T .v/). So the output is a parallelogram. Problem Set 7.2, page 395 2 0 For S v D d 2 v=dx 2 60 2 3 1 v 1 , v 2 , v 3 , v 4 D 1, x , x , x The matrix for S is B D 4 0 Sv1 D Sv2 D 0, Sv3 D 2v1 , Sv4 D 6v2 ; 0 0 0 0 0 2 0 0 0 3 0 67 . 05 0 2 S v D d 2 v=dx 2 D 0 for linear functions v.x / D a C bx . All .a; b; 0; 0/ are in the nullspace of the second derivative matrix B . 3 (Matrix A)2 D B when (transformation T )2 D S and output basis = i...
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