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Mathematics Chapter 1 For the exclusive use of adopters of the book
Solutions — Chapter 1 1.1.1. (a) Reduce the system to x−y= 7,3y=−4; then use Back Substitution to solve for x=17 3, y =−4 3. (b) Reduce the system to 6u+v= 5,−5 2v=5 2; then use Back Substitution to solve for […]
Mathematics Chapter 1 Running Gaussian Elimination with pivoting
1.7.19. (a)x=−220., y = 26, z =.91; (b)x=−190., y = 24, z =.84; (c)x=−210, y= 26, z = 1.(d) The exact solution is x=−213.658, y = 25.6537, z =.858586. Full pivoting is the most accurate. Interestingly, partial pivoting fares a […]
Mathematics Chapter 1 The effect of multiplying P AP Tis equivalent to
1.5.26. Applying Gaussian Elimination: E1=0 @1 0 −1 √311 A, E1A=0 B @ √3 2−1 2 02 √3 1 C A, 1.5.27. (a)0 @−i 2 1 2 1 2−i 21 A, (b) −1 1 −i 1 + i −1!, (c)0 […]
Mathematics Chapter 10 That Tis complete with a single dominant
10.4.6. The transition matrix T=0 B B B @ 02 32 3 1 0 1 3 01 30 1 C C C Ais regular because T4=0 B B B @ 14 27 26 81 26 81 2 949 81 16 […]
Mathematics Chapter 10 Then the population will increase without limit
Solutions — Chapter 10 10.1.1. (a)u(1) = 2, u(10) = 1024, u(20) = 1048576; unstable. 10.1.2. (a)u(k+1) = 1.0325 u(k),u(0) = 100, where u(k)represents the balance after kyears. (b)u(10) = 1.032510 ×100 = 137.69 dollars. (c)u(k+1) = (1 + .0325/12) […]
Mathematics Chapter 11 The same ideas apply to clamped splines
11.4.14. (a)u(x) = 8 < :−1.25(x+ 1)3+ 4.25(x+ 1) −2,−1≤x≤0, 1.25×3−3.75×2+.5x−1,0≤x≤1. -1 -0.5 0.5 1 -2 -1.5 -1 -0.5 0.5 1 (b)u(x) = 8 > > > < > > > : −x3+ 2x+ 1,0≤x≤1, 2(x−1)3−3(x−1)2−(x−1) + 2,1≤x≤2, −(x−2)3+ 3(x−2)2−(x−2),2≤x≤3. […]
Mathematics Chapter 11 We conclude that the sequence satisfies
Solutions — Chapter 11 11.1.1. The greatest displacement is at x=d α+1 2, with u(x) = α 8+d 2+d2 2a, when d < 2α, and at x= 1, with u(x) = d, when d≥2α. The greatest stress and greatest strain […]
Mathematics Chapter 2 Associativity of Scalar Multiplication
Solutions — Chapter 2 2.1.1. Commutativity of Addition: (x+ i y) + (u+ i v) = (x+u) + i (y+v) = (u+ i v) + (x+ i y). Associativity of Addition: (x+ i y) + h(u+ i v) + (p+ […]
Mathematics Chapter 2 Thus The Matrices Agree Basis Which Enough
then choose another vector vm+2 not in the span of v1, . . . , vm+1, and so v1, . . . , vm+2 are also linearly independent. We continue on in this fashion until we arrive at nlin- early […]
Mathematics Chapter 3 A Gram matrix is positive definite if
3.3.32. (a)v= ( a, 0 )Tor ( 0, a )T; (b)v= ( a, 0 )Tor ( 0, a )T; (c)v= ( a, 0 )Tor ( 0, a )T; (d)v= ( a, a )Tor ( a, −a)T. 3.3.33. Let 0 < […]
Mathematics Chapter 3 At least one of which is strictly positive
Solutions — Chapter 3 3.1.1. Bilinearity: hcu+dv,wi= (cu1+dv1)w1−(cu1+d v1)w2−(c u2+d v2)w1+b(c u2+dv2)w2 =c(u1w1−u1w2−u2w1+bu2w2) + d(v1w1−v1w2−v2w1+bv2w2) =chu,wi+dhv,wi, 3.1.2. (a), (f) and (g) define inner products; the others don’t. 3.1.3. It is not positive definite, since if v= ( 1,−1 )T, say, […]
Mathematics Chapter 4 The maximal distance between the point
Solutions — Chapter 4 4.1.1. We need to minimize (3x−1)2+ (2 x+ 1)2= 13 x2−2x+ 2. The minimum value of 25 13 occurs when x=1 13 . 4.1.2. Note that f(x, y)≥0; the minimum value f(x?, y?) = 0 is […]
Mathematics Chapter 5 Is orthogonal to all polynomials of degree
-1 -0.5 0.5 1 -1.5 -1 -0.5 0.5 1 1.5 -1 -0.5 0.5 1 -1.5 -1 -0.5 0.5 1 1.5 -1 -0.5 0.5 1 -1.5 -1 -0.5 0.5 1 1.5 5.4.21. The Gram–Schmidt process will lead to the monic Chebyshev […]
Mathematics Chapter 5 The three vectors are nonzero, mutually orthogonal
Solutions — Chapter 5 5.1.1. (a) Orthogonal basis; (b) orthonormal basis; (c) not a basis; (d) basis; (e) orthogonal basis; (f) orthonormal basis. 5.1.2. (a) Basis; (b) orthonormal basis; (c) not a basis. 5.1.3. (a) Basis; (b) basis; (c) not […]
Mathematics Chapter 6 For maximum displacement of the bottom mass
Solutions — Chapter 6 6.1.1. (a)K= 3−2 −2 3 !; (b)u=0 @ 18 5 17 51 A= 3.6 3.4!; (c) the first mass has moved the farthest; (d)e=“18 5,−1 5,−17 5”T= ( 3.6,−.2,−3.4 )T, so the first spring has stretched […]
Mathematics Chapter 7 And represent either a reflection through the origin
with 0 @ 3 23 2 1 21 21 A= 1 2 0 1 !0 @ 1 21 2 1 21 21 A, 2 2!= 1 2 0 1 ! 1 0!+ 1 2!; (d)T6◦T3[x] = 0 @ 1 23 […]
Mathematics Chapter 7 The linear function exists and is unique if
Solutions — Chapter 7 7.1.1. Only (a) and (d) are linear. 7.1.3. (a)F(0,0) = 2 0!6= 0 0!, (b)F(2x, 2y) = 4F(x, y)6= 2F(x, y), (c)F(−x, −y) = F(x, y)6=−F(x, y), (d)F(2x, 2y)6= 2F(x, y), (e)F(0,0) = 1 0!6= 0 […]
Mathematics Chapter 8 The equilibrium solution is asymptotically stable
Solutions — Chapter 8 8.1.1. (a)u(t) = −3e5t, (b)u(t) = 3e2(t−1), (c)u(t) = e−3(t+1). 8.1.2. γ= log 2/100 ≈.0069. After 10 years: 93.3033 gram; after 100 years: 50 gram; after 1000 years: .0977 gram. 8.1.6. The solution is u(t) = […]
Mathematics Chapter 8 where Λ is the diagonal eigenvalue matrix
(c) Eigenvalues: 7,1; eigenvectors: 1 2 1!, 1 0!. B 6 C B C B 2 C ♦8.4.10. If L[v] = λv, then, using the inner product, λkvk2=hL[v],vi=hv, L[v]i=λkvk2, which proves that the eigenvalue λis real. Similarly, if L[w] = […]
Mathematics Chapter 9 Equating this to the right hand side of the second
Solutions — Chapter 9 9.1.1. (i) (a)u(t) = c1cos 2t+c2sin 2t. (b)du dt = 0 1 −4 0 !u. (c)u(t) = c1cos 2t+c2sin 2t −2c1sin 2t+ 2c2cos 2t!. (d)(e) 246 8 0.2 0.4 (iii ) (a)u(t) = c1e−t+c2te−t. (b)du dt […]
Mathematics Chapter 9 Oder of the springs does not change the frequencies
9.4.37. (a)0 B @ e2t0 0 0et0 0 0 1 1 C A— scalings by a factor λ=etin the ydirection and λ2=e2tin the x direction. The trajectories are the semi-parabolas x=c y2, z =dfor c, d constant, and the half-lines […]