Linear Algebra Exercises-n-Answers.pdf
Linear Algebra Exercises-n-Answers.pdf Linear Algebra Exercises-n-Answers.pdf
24 Linear Algebra, by Hefferon and then as required. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⃗u ( k⃗v + m ⃗w ) u 1 ⎜ = ⎝ ⎟ . ⎠ ( kv 1 mw 1 ⎜ ⎝ ⎟ ⎜ . ⎠ + ⎝ ⎟ . ⎠ ) u n kv n mw n ⎛ ⎞ ⎛ ⎞ u 1 kv 1 + mw 1 ⎜ ⎟ ⎜ ⎟ = ⎝ . ⎠ ⎝ . ⎠ u n kv n + mw n = u 1 (kv 1 + mw 1 ) + · · · + u n (kv n + mw n ) = ku 1 v 1 + mu 1 w 1 + · · · + ku n v n + mu n w n = (ku 1 v 1 + · · · + ku n v n ) + (mu 1 w 1 + · · · + mu n w n ) = k(⃗u ⃗v) + m(⃗u ⃗w) One.II.2.38 For x, y ∈ R + , set ⃗u = (√ ) x √ y ⃗v = (√ ) √ y x so that the Cauchy-Schwartz inequality asserts that (after squaring) as desired. ( √ x √ y + √ y √ x) 2 ≤ ( √ x √ x + √ y √ y)( √ y √ y + √ x √ x) (2 √ x √ y) 2 ≤ (x + y) 2 √ xy ≤ x + y 2 One.II.2.39 This is how the answer was given in the cited source. The actual velocity ⃗v of the wind is the sum of the ship’s velocity and the apparent velocity of the wind. Without loss of generality we may assume ⃗a and ⃗ b to be unit vectors, and may write ⃗v = ⃗v 1 + s⃗a = ⃗v 2 + t ⃗ b where s and t are undetermined scalars. Take the dot product first by ⃗a and then by ⃗ b to obtain s − t⃗a ⃗ b = ⃗a (⃗v 2 − ⃗v 1 ) s⃗a ⃗ b − t = ⃗ b (⃗v 2 − ⃗v 1 ) Multiply the second by ⃗a ⃗ b, subtract the result from the first, and find s = [⃗a − (⃗a ⃗ b) ⃗ b] (⃗v 2 − ⃗v 1 ) 1 − (⃗a ⃗ b) 2 . Substituting in the original displayed equation, we get One.II.2.40 We use induction on n. In the n = 1 base case the identity reduces to and clearly holds. ⃗v = ⃗v 1 + [⃗a − (⃗a ⃗ b) ⃗ b] (⃗v 2 − ⃗v 1 )⃗a 1 − (⃗a ⃗ b) 2 . (a 1 b 1 ) 2 = (a 1 2 )(b 1 2 ) − 0 For the inductive step assume that the formula holds for the 0, . . . , n cases. We will show that it
Answers to Exercises 25 then holds in the n + 1 case. Start with the right-hand side ( ∑ 2 a )( ∑ 2 j b ) ∑ ( ) 2 j − ak b j − a j b k 1≤j≤n+1 1≤j≤n+1 1≤j≤n 1≤k
- 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: 22 Linear Algebra, by Hefferon One.
- Page 28 and 29: 26 Linear Algebra, by Hefferon One.
- 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: 58 Linear Algebra, by Hefferon and
- Page 62 and 63: 60 Linear Algebra, by Hefferon Two.
- Page 64 and 65: 62 Linear Algebra, by Hefferon Two.
- Page 66 and 67: 64 Linear Algebra, by Hefferon Two.
- 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
<strong>Answers</strong> to <strong>Exercises</strong> 25<br />
then holds in the n + 1 case. Start with the right-hand side<br />
( ∑<br />
2<br />
a )( ∑<br />
2<br />
j b ) ∑ ( ) 2<br />
j −<br />
ak b j − a j b k<br />
1≤j≤n+1 1≤j≤n+1<br />
1≤j≤n<br />
1≤k
Change language