This preview shows page 1. Sign up to view the full content.
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 leftright) (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...
View
Full
Document
 Spring '12
 Minki
 Mass

Click to edit the document details