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About 3307 results. 1294 free access solutions
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| m55201 | Prove that if Q is nonsingular matrix with Qt = Q−1, then Q is orthogonal. |
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| m55206 | Prove that the following sequences are convergent, and find their limits.
a. x(k) = (1/k, e1−k ,−2/k2)t
b. x(k) = (e−k cos k, k sin(1/k), 3 + k )t
c. x(k) = (ke−k2 , (cos k)/k, √(k2 + k) − k)t
d. x(k) = (e1/k , (k2 + 1)/(1 − k2), (1/k2)(1 + 3 + 5+· · ·+(2k − 1)))t |
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| m55210 | Prove that the set of vectors {v1, v2 , . . . , vk} described in the Gram-Schmidt Theorem is orthogonal. |
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| m55212 | Prove the following statements or provide counterexamples to show they are not true.
a. The product of two symmetric matrices is symmetric.
b. The inverse of a nonsingular symmetric matrix is a nonsingular symmetric matrix.
c. If A and B are n × n matrices, then (AB)t = AtBt . |
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| m55217 | Prove the Generalized Rolle s Theorem, Theorem 1.10, by verifying the following.
a. Use Rolle s Theorem to show that f (zi) = 0 for n − 1 numbers in [a, b] with a < z1 < z2 < · · · < zn−1 < b.
b. Use Rolle s Theorem to show that f (wi) = 0 for n−2 numbers in [a, b] with z1 < w1 < z2 < w2 · · ·wn−2 < zn−1 < b.
c. Continue the arguments in a. and b. to show that for each j = 1, 2. . . n − 1 there are n - j distinct numbers in [a, b] where f(j) is 0.
d. Show that part c. implies the conclusion of the theorem. |
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| m55222 | Prove the statement following Definition 4.1; that is, show that a quadrature formula has degree of precision n if and only if the error E(P(x)) = 0 for all polynomials P(x) of degree k = 0, 1, . . . , n, but E(P(x)) ≠ 0 for some polynomial P(x) of degree n + 1. |
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| m55226 | Prove Theorem 11.3. |
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| m55228 | Prove Theorem 5.3 by applying the Mean Value Theorem 1,8 to f (t, y), holding t fixed. |
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| m55230 | Prove Theorem 6.31. [Show that [ui,i+1] < 1, for each i = 1, 2, . . . , n − 1, and that |lii| > 0, for each i = 1, 2, . . . , n. Deduce that det A = det L · det U ≠ 0.] |
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| m55231 | Prove Theorem 7.33 using mathematical induction as follows:
a. Show that (r(1), v(1)) = 0.
b. Assume that (r(k), v(j)) = 0, for . each k ≤ l and j = 1, 2, . . . , k, and show that this implies that (r(l+1), v(j)) = 0, for each j = 1, 2, . . . , l.
c. Show that (r(l+1), v(l+1)) = 0. |
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| m55232 | Prove Theorem 9.10.
Theorem 9.10
Suppose that Q is an orthogonal n × n matrix. Then
(i) Q is invertible with Q−1 = Qt;
(ii) For any x and y in Rn, (Qx)t Qy = xty;
In addition, the converse of part (i) holds. That is,
• any invertible matrix Q with Q−1 = Qt is orthogonal.
As an example, the permutation matrices discussed in Section 6.5 have this property, so they are orthogonal. |
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| m55233 | Prove Theorem 9.13.
Theorem 9.13
An n × n matrix A is similar to a diagonal matrix D if and only if A has n linearly independent eigenvectors. In this case, D = S−1 AS, where the columns of S consist of the eigenvectors, and the ith diagonal element of D is the eigenvalue of A that corresponds to the ith column of S.
The pair of matrices S and D is not unique. For example, any reordering of the columns of S and corresponding reordering of the diagonal elements of D will give a distinct pair.
We saw in Theorem 9.3 that the eigenvectors of a matrix that correspond to distinct eigenvalues form a linearly independent set. |
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| m55299 | Repeat Exercise 1 using ω = 1.3.
In Exercise 1
a. 3x1 − x2 + x3 = 1,
3x1 + 6x2 + 2x3 = 0,
3x1 + 3x2 + 7x3 = 4.
b. 10x1 − x2 = 9,
−x1 + 10x2 − 2x3 = 7,
− 2x2 + 10x3 = 6.
c. 10x1 + 5x2 = 6,
5x1 + 10x2 − 4x3 = 25,
− 4x2 + 8x3 − x4 = −11,
− x3 + 5x4 = −11.
d. 4x1 + x2 + x3 + x5 = 6,
−x1 − 3x2 + x3 + x4 = 6,
2x1 + x2 + 5x3 − x4 − x5 = 6,
−x1 − x2 − x3 + 4x4 = 6,
2x2 − x3 + x4 + 4x5 = 6 |
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| m55300 | Repeat Exercise 1 using Algorithm 6.2.
In Exercise 1
a. x1 − 5x2 + x3 = 7,
10x1 + 20x3 = 6,
5x1 − x3 = 4
b. x1 + x2 − x3 = 1,
x1 + x2 + 4x3 = 2,
2x1 − x2 + 2x3 = 3
c. 2x1 − 3x2 + 2x3 = 5,
−4x1 + 2x2 − 6x3 = 14,
2x1 + 2x2 + 4x3 = 8.
d. x2 + x3 = 6,
x1 − 2x2 − x3 = 4,
x1 − x2 + x3 = 5. |
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| m55301 | Repeat Exercise 1 using Algorithm 6.3.
In Exercise 1
a. x1 − 5x2 + x3 = 7,
10x1 + 20x3 = 6,
5x1 − x3 = 4
b. x1 + x2 − x3 = 1,
x1 + x2 + 4x3 = 2,
2x1 − x2 + 2x3 = 3
c. 2x1 − 3x2 + 2x3 = 5,
−4x1 + 2x2 − 6x3 = 14,
2x1 + 2x2 + 4x3 = 8.
d. x2 + x3 = 6,
x1 − 2x2 − x3 = 4,
x1 − x2 + x3 = 5. |
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| m55302 | Repeat Exercise 1 using complete pivoting.
In Exercise 1
a. x1 − 5x2 + x3 = 7,
10x1 + 20x3 = 6,
5x1 − x3 = 4
b. x1 + x2 − x3 = 1,
x1 + x2 + 4x3 = 2,
2x1 − x2 + 2x3 = 3
c. 2x1 − 3x2 + 2x3 = 5,
−4x1 + 2x2 − 6x3 = 14,
2x1 + 2x2 + 4x3 = 8.
d. x2 + x3 = 6,
x1 − 2x2 − x3 = 4,
x1 − x2 + x3 = 5. |
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| m55303 | Repeat Exercise 1 using cubic Bézier polynomials.
In Exercise 1
Let (x0, y0) = (0, 0) and (x1, y1) = (5, 2) be the endpoints of a curve. Use the given guidepoints to construct parametric cubic Hermite approximations (x(t), y(t)) to the curve, and graph the approximations.
a. (1, 1) and (6, 1)
b. (0.5, 0.5) and (5.5, 1.5)
c. (1, 1) and (6, 3)
d. (2, 2) and (7, 0) |
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| m55304 | Repeat Exercise 1 using four-digit rounding arithmetic, and compare the errors to those in Exercise 3.
In Exercise 1
a.
b. |
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| m55305 | Repeat Exercise 1 using four-digit rounding arithmetic.
In Exercise 1
a. f (x) = ln x, x0 = 1.0, h = 0.4
b. f (x) = x + ex , x0 = 0.0, h = 0.4
c. f (x) = 2x sin x, x0 = 1.05, h = 0.4
d. f (x) = x3 cos x, x0 = 2.3, h = 0.4 |
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| m55306 | Repeat Exercise 1 using Heun s method.
In Exercise 1 |
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