Linear Algebra Exercises-n-Answers.pdf

More documents

Recommendations

Info

130 Linear Algebra, by Hefferon then the vector ⃗w whose components are v 2 , −v 1 , 0, . . . , 0 is perpendicular to ⃗v. In either case, observe that ⃗v + ⃗w does not equal ⃗v, and that ⃗v is the projection of ⃗v + ⃗w into l. (⃗v + ⃗w) ⃗v · ⃗v = ( ⃗v ⃗v ⃗w ⃗v ) ⃗v ⃗v + · ⃗v = · ⃗v = ⃗v ⃗v ⃗v ⃗v ⃗v ⃗v ⃗v ⃗v ⃗v We can dispose of the remaining n = 0 and n = 1 cases. The dimension n = 0 case is the trivial vector space, here there is only one vector and so it cannot be expressed as the projection of a different vector. In the dimension n = 1 case there is only one (nondegenerate) line, and every vector is in it, hence every vector is the projection only of itself. Three.VI.1.15 The proof is simply a calculation. ⃗v ⃗s ⃗s |⃗v ⃗s | |⃗v ⃗s | ‖ · ⃗s ‖ = |⃗v | · ‖⃗s ‖ = · ‖⃗s ‖ = ⃗s ⃗s ⃗s ⃗s ‖⃗s ‖ 2 ‖⃗s ‖ Three.VI.1.16 Because the projection of ⃗v into the line spanned by ⃗s is ⃗v ⃗s ⃗s ⃗s · ⃗s the distance squared from the point to the line is this (a vector dotted with itself ⃗w ⃗w is written ⃗w 2 ). ⃗v ⃗s ‖⃗v − ⃗s ⃗s · ⃗s ⃗v ⃗s ⃗s ⃗s ‖2 = ⃗v ⃗v − ⃗v ( · ⃗s) − (⃗v · ⃗s ) ⃗v + (⃗v · ⃗s )2 ⃗s ⃗s ⃗s ⃗s ⃗s ⃗s ⃗v ⃗s ⃗s = ⃗v ⃗v − 2 · ( ) · ⃗v ⃗s + (⃗v ) · ⃗s ⃗s ⃗s ⃗s ⃗s ⃗s = (⃗v ⃗v ) · (⃗s ⃗s ) − 2 · (⃗v ⃗s )2 + (⃗v ⃗s ) 2 ⃗s ⃗s (⃗v ⃗v )(⃗s ⃗s ) − (⃗v ⃗s )2 = ⃗s ⃗s Three.VI.1.17 Because square root is a strictly increasing function, we can minimize d(c) = (cs 1 − v 1 ) 2 +(cs 2 −v 2 ) 2 instead of the square root of d. The derivative is dd/dc = 2(cs 1 −v 1 )·s 1 +2(cs 2 −v 2 )·s 2 . Setting it equal to zero 2(cs 1 − v 1 ) · s 1 + 2(cs 2 − v 2 ) · s 2 = c · (2s 2 1 + 2s 2 2) − (v 1 s 1 + v 2 s 2 ) = 0 gives the only critical point. c = v 1s 1 + v 2 s 2 ⃗v ⃗s = s 12 + s 2 2 ⃗s ⃗s Now the second derivative with respect to c d 2 d dc 2 = 2s 1 2 2 + 2s 2 is strictly positive (as long as neither s 1 nor s 2 is zero, in which case the question is trivial) and so the critical point is a minimum. The generalization to R n is straightforward. Consider d n (c) = (cs 1 − v 1 ) 2 + · · · + (cs n − v n ) 2 , take the derivative, etc. Three.VI.1.18 The Cauchy-Schwartz inequality |⃗v ⃗s | ≤ ‖⃗v ‖ · ‖⃗s ‖ gives that this fraction ⃗v ⃗s ⃗s |⃗v ⃗s | |⃗v ⃗s | ‖ · ⃗s ‖ = |⃗v | · ‖⃗s ‖ = · ‖⃗s ‖ = ⃗s ⃗s ⃗s ⃗s ‖⃗s ‖ 2 ‖⃗s ‖ when divided by ‖⃗v ‖ is less than or equal to one. That is, ‖⃗v ‖ is larger than or equal to the fraction. Three.VI.1.19 Write c⃗s for ⃗q, and calculate: (⃗v c⃗s/c⃗s c⃗s ) · c⃗s = (⃗v ⃗s/⃗s ⃗s ) · ⃗s. Three.VI.1.20 (a) Fixing ( 1 ⃗s = 1) as the vector whose span is the line, the formula gives this action, ( ) ( x 1 ( ( x y 1) 1 ↦→ ( ) ( · = y) 1 1 1) 1 1) x + y ( ( ) 1 (x + y)/2 · = 2 1) (x + y)/2 which is the effect of this matrix. ( 1/2 ) 1/2 1/2 1/2 (b) Rotating the entire plane π/4 radians clockwise brings the y = x line to lie on the x-axis. Now projecting and then rotating back has the desired effect.
Answers to Exercises 131 Three.VI.1.21 the projections are these. The sequence need not settle down. With ( ( ) 1 ⃗a = ⃗ 1 b = 0) 1 ⃗v 1 = This sequence doesn’t repeat. ( ) 1/2 , ⃗v 1/2 2 = ( ) 1/2 , ⃗v 0 3 = ( ) 1/4 , . . . 1/4 Subsection Three.VI.2: Gram-Schmidt Orthogonalization Three.VI.2.9 (a) ( 1 ⃗κ 1 = 1) ( ) ( 2 1 ( ( ( ( 2 2 2 1 1) 1 ⃗κ 2 = − proj 1) [⃗κ1]( ) = − ) ( · = 1) 1) 1 1) 1) ( 1 1 ( 2 − 1) 3 ( ( ) 1 1/2 2 1) · = −1/2 (b) ( 0 ⃗κ 1 = 1) ⃗κ 2 = ( ) −1 − proj 3 [⃗κ1 ]( ( −1 3 ) ) = ( ) −1 − 3 ( ) ( −1 0 ( 3 1) 0 ( ) ( · = 0 0 1) 1 1) ( ) −1 − 3 ( 0 3 1 1) · = ( ) −1 0 (c) ( 0 ⃗κ 1 = 1) ⃗κ 2 = ( ) −1 − proj 0 [⃗κ1 ]( ( −1 0 ) ) = ( ) −1 − 0 ( ) ( −1 0 ( 0 1) 0 ( ) ( · = 0 0 1) 1 1) ( ) −1 − 0 ( 0 0 1 1) · = ( ) −1 0 The corresponding orthonormal bases for the three parts of this question are these. ( √ ) ( √ ) ( ( ) ( ( ) 1/ 2 〈 1/ √ 2/2 , 2 − √ 0 −1 0 −1 〉〈 , 〉〈 , 〉 2/2 1) 0 1) 0
Page 1 and 2:
Answers to Exercises Linear Algebra
Page 3:
These are answers to the exercises
Page 6 and 7:
4 Linear Algebra, by Hefferon Topic
Page 8 and 9:
6 Linear Algebra, by Hefferon One.I
Page 10 and 11:
8 Linear Algebra, by Hefferon (this
Page 12 and 13:
10 Linear Algebra, by Hefferon from
Page 14 and 15:
12 Linear Algebra, by Hefferon (d)
Page 16 and 17:
14 Linear Algebra, by Hefferon One.
Page 18 and 19:
16 Linear Algebra, by Hefferon so t
Page 20 and 21:
18 Linear Algebra, by Hefferon ⎛
Page 22 and 23:
20 Linear Algebra, by Hefferon inst
Page 24 and 25:
Page 26 and 27:
24 Linear Algebra, by Hefferon and
Page 28 and 29:
Page 30 and 31:
28 Linear Algebra, by Hefferon (a)
Page 32 and 33:
30 Linear Algebra, by Hefferon The
Page 34 and 35:
32 Linear Algebra, by Hefferon > A:
Page 36 and 37:
34 Linear Algebra, by Hefferon Topi
Page 38 and 39:
36 Linear Algebra, by Hefferon 65 5
Page 40 and 41:
38 Linear Algebra, by Hefferon Two.
Page 42 and 43:
40 Linear Algebra, by Hefferon oute
Page 44 and 45:
42 Linear Algebra, by Hefferon (e)
Page 46 and 47:
44 Linear Algebra, by Hefferon Asso
Page 48 and 49:
46 Linear Algebra, by Hefferon of B
Page 50 and 51:
48 Linear Algebra, by Hefferon (c)
Page 52 and 53:
50 Linear Algebra, by Hefferon Poss
Page 54 and 55:
52 Linear Algebra, by Hefferon Repe
Page 56 and 57:
54 Linear Algebra, by Hefferon give
Page 58 and 59:
56 Linear Algebra, by Hefferon (b)
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
66 Linear Algebra, by Hefferon But
Page 70 and 71:
68 Linear Algebra, by Hefferon (b)
Page 72 and 73:
70 Linear Algebra, by Hefferon 2 Fo
Page 74 and 75:
72 Linear Algebra, by Hefferon -a T
Page 76 and 77:
74 Linear Algebra, by Hefferon 3 (a
Page 78 and 79:
Page 80 and 81:
78 Linear Algebra, by Hefferon (d)
Page 82 and 83: 80 Linear Algebra, by Hefferon Thre
Page 84 and 85: 82 Linear Algebra, by Hefferon has
Page 86 and 87: 84 Linear Algebra, by Hefferon (c)
Page 92 and 93: 90 Linear Algebra, by Hefferon ∣
Page 98 and 99: 96 Linear Algebra, by Hefferon and,
Page 100 and 101: 98 Linear Algebra, by Hefferon The
Page 104 and 105: 102 Linear Algebra, by Hefferon mea
Page 108 and 109: 106 Linear Algebra, by Hefferon Thr
Page 110 and 111: 108 Linear Algebra, by Hefferon (e)
Page 112 and 113: 110 Linear Algebra, by Hefferon (a)
Page 114 and 115: 112 Linear Algebra, by Hefferon to
Page 120 and 121: 118 Linear Algebra, by Hefferon As
Page 122 and 123: 120 Linear Algebra, by Hefferon and
Page 124 and 125: 122 Linear Algebra, by Hefferon we
Page 126 and 127: 124 Linear Algebra, by Hefferon (b)
Page 128 and 129: 126 Linear Algebra, by Hefferon As
Page 130 and 131: 128 Linear Algebra, by Hefferon (d)
Page 136 and 137: and that (⃗v − (c1 · ⃗κ 1 +
Page 138 and 139: 136 Linear Algebra, by Hefferon (a)
Page 140 and 141: 138 Linear Algebra, by Hefferon the
Page 146 and 147: 144 Linear Algebra, by Hefferon Pro
Page 148 and 149: 146 Linear Algebra, by Hefferon 5 T
Page 150 and 151: 148 Linear Algebra, by Hefferon Top
Page 152 and 153: 150 Linear Algebra, by Hefferon 4 R
Page 154 and 155: 152 Linear Algebra, by Hefferon > 0
Page 156 and 157: 154 Linear Algebra, by Hefferon 0.5
Page 158 and 159: 156 Linear Algebra, by Hefferon n =
Page 160 and 161: 158 Linear Algebra, by Hefferon Top
Page 166 and 167: 164 Linear Algebra, by Hefferon Fou
Page 170 and 171: 168 Linear Algebra, by Hefferon The
Page 176 and 177: 174 Linear Algebra, by Hefferon whe
Page 178 and 179: 176 Linear Algebra, by Hefferon 0.2
Page 180 and 181: 178 Linear Algebra, by Hefferon Als
Page 182 and 183:
180 Linear Algebra, by Hefferon Fiv
Page 184 and 185:
182 Linear Algebra, by Hefferon Res
Page 186 and 187:
184 Linear Algebra, by Hefferon The
Page 188 and 189:
186 Linear Algebra, by Hefferon (as
Page 190 and 191:
Page 192 and 193:
190 Linear Algebra, by Hefferon to
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
196 Linear Algebra, by Hefferon tha
Page 200 and 201:
198 Linear Algebra, by Hefferon Aga
Page 202 and 203:
200 Linear Algebra, by Hefferon Bec
Page 204 and 205:
202 Linear Algebra, by Hefferon (c)
Page 206 and 207:
Page 208 and 209:
206 Linear Algebra, by Hefferon For
Page 210 and 211:
Page 212 and 213:
show all

Linear Algebra Exercises-n-Answers.pdf

Create successful ePaper yourself

Delete template?

Save as template?