| № |
Condition |
free/or 0.5$ |
| m96578 | Solve Exercise 2 using QR-factorization of A
In Exercise 2 |
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| m96579 | Solve for a, b, c, d if |
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| m96580 | Solve for x in each of the following.
(a) log10 x + log10 6 = 1
(b) In x - ln(x - 1) = In 3
(c) log3(x2 + 4x + 4) - log3(2x - 5) = 2 |
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| m96582 | Solve the following linear congruences for x.
(a) 3x = 1 (mod 31)
(b) 5x = 8 (mod 37)
(c) 6x = 97 (mod 125) |
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| m96583 | Solve the following systems of linear equations by using matrices:
(a) 3x - 2y = 5
4x - 3y = 6
(b) 5x + 3y = 35
3x - 2y = 2 |
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| m96584 | Solve the linear system, with the given augmented matrix, if it is consistent.
(a)
(b) |
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| m96599 | State a result that generalizes the ideas presented in the previous two exercises. |
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| m96602 | State the dual of each theorem in Exercise 13. (Here you will want to use the result of Example 3.19 in conjunction with Theorem 3.5.)
In Exercise 13
Consider the membership table (Table 3.7). If we are given the condition that A ⊂ B, then we need consider only those rows of the table for which this is true - rows 1, 2, and 4, as indicated by the arrows. For these rows, the columns for B and A U B are exactly the same, so this membership table shows that A ⊂ B ⇒ A U B = B.
Table 3.7
Use membership tables to verify each of the following:
a) A ⊂ B ⇒ A ∩ B = A
b) [(A ∩ B = A) ∧ (B U C = C)] ⇒ A U B U C = C
c)
d) A ∆ B = C ⇒ A ∆ C = B and B ∆ C = A |
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| m96629 | Suppose S" is a statement about n for each n ≥ 1. Explain what must be done to prove that Sn is true for all n ≥ 1 if it is known that
(a) Sn ⇒ Sn+2 for each n ≥ 1.
(b) Sn ⇒ Sn+8 for each n ≥ 1.
(c) Sn ⇒ Sn+1 for each n ≥ 10.
(d) Both Sn and Sn+i ⇒ Sn+2 for each n ≥ 1. |
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| m96630 | Suppose T: P2 → R2 is a linear transformation. If B = {1, x, x2} and D = {(1, 1), (0, 1)}, find the action of T given:
(a )MDB(T) =
(b) MDB(T) = |
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| m96637 | Suppose that 5 = {v1, v2, v3} is a linearly independent set of vectors in a vector space V. Is T = {w1, w2, w3}, where w1 = v1 + v2, w2 = v1 + v3, w3 = v2 + v3, linearly dependent or linearly independent? Justify your answer. |
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| m96640 | Suppose that A, B are independent with Pr(A U B) = 0.6 and Pr(A) = 0.3. Find Pr(B). |
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| m96641 | Suppose that a, b, c ∈ Z and 5|(a2 + b2 + c2). Prove that 5|a or 5 or 5|c. |
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| m96643 | Suppose that a, b, k ∈ Z+ with a - b - p1e1p2e2 ∙∙∙∙∙∙∙∙ pkek, for P1, P2, ∙∙∙∙∙∙∙∙ Pk prime and e1, e2, . .. , ek ∈ Z+. For how many values of n ( > 1) is a = b (mod n) true? |
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| m96653 | Suppose that A is an n ( n matrix and that there is no nonzero vector x in Rn such that Ax = x. Show that A - In is nonsingular. |
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| m96657 | Suppose that a random variable Z has mean E(X) = 17 and variance Var(X) = 9, but its probability distribution is unknown. Use Chebyshev s Inequality to estimate a lower bound for
(a) Pr(11 < Z < 23).
(b) Pr(10 < Z < 24).
(c) Pr(8 < Z < 26). |
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| m96658 | Suppose that a random variable Z has mean E(X) = 15 and variance Var(X) = 4, but its probability distribution is unknown. Use Chebyshev s Inequality to find the value of the constant c where Pr(|X - 15| < c) > 0.96. |
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| m96665 | Suppose that an airplane is flying with an air speed of 260 kilometers per hour while a wind is blowing to the west at 100 kilometers per hour. Indicate on a figure the appropriate direction that the plane must follow to fly directly south. What will be the resultant speed? |
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| m96691 | Suppose that f: B3 → B is defined by
(a) Determine the d.n.f. and c.n.f. for f.
(b) Write f as a sum of minterms and as a product of maxterms (utilizing binary labels). |
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| m96709 | Suppose that G = (V, E) is a loop-free planar graph with | V | = v, | E | = e, and k (G) = the number of components of G.
(a) State and prove an extension of Euler s Theorem for such a graph,
(b) Prove that Corollary 11.3 remains valid if G is loop-free and planar but not connected. |
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