3. Use type vectors to establish the bijection (mentioned in the proof of Theorem 2.4.1) between ...
(complete the proof. Hint: Use the Squeeze Theorem to show that lima = L.) 3- For all ne N, let an = Let S = {a, neN). 3-1) Use the fact that lim 0 and the result of Exercise 1 to show that OES'. 3-2) Use the result of Exercise 2 to show that S - {0}. 4- Prove that
1Hint: Use the theorem from class that any linearly independent list of vectors is contained in a basis 2Hint: Remember that we prove the equality of sets X = Y by showing X ⊂ Y and Y ⊂ X. (2 points each for (a),(b),(d)) In this problem, we will prove the following di- mension formula. Theorem. If H and H' are subspaces of a finite-dimensional vector space V, then dim(H+H') = dim(H)+dim(H') - dim(H nH'). (a) Suppose {u1;...; up} is...
QUESTION: PROVE THE FOLLOWING 4.3 THEOREM IN THE CASE r=1(no induction required, just use the definition of the determinants) Theorem 4.3. The determinant of an n × n matrix is a linear function of each row when the remaining rows are held fixed. That is, for 1 Sr S n, we have ar-1 ar-1 ar-1 ar+1 ar+1 ar+1 an an rt whenever k is a scalar and u, v, and each a are row vectors in F". Proof. The proof...
9. Use the construction in the proof of the Chinese remainder theorem to find a solution to the system of congruences X 1 mod 2 x 2 mod 3 x 3 mod 5 x 4 mod 11 10. Use Fermats little theorem to find 712 mod 13 11. What sequence of pseudorandom numbers is generated using the linear congruential generator Xn+1 (4xn + 1) mod 7 with seed xo 3? 9. Use the construction in the proof of the Chinese...
4. 20 points (Ex 21.2-3 of text book) Adapt the aggregate proof of Theorem 21.1 in the text book (slide 44 of Lecture5) to obtain amortized time bounds of O(1) for Make-Set and Find-Set and O(log n) for Union using linked list representation and weighted-union heuristic. Theorem 21.1 Using the linked-list representation of disjoint sets and the weighted-union heuris- tic, a sequence of m MAKE-SET, UNION, and FIND-SET operations, n of which are MAKE-SET operations, takes O(m + n lgn)...
Use the Taylor Remainder theorem to find the smallest value of n such that Rn (3 when f(x) = -2 1000 on (-4, 4) with a = = 0. 1
Let n > 1, and let S = {1, 2, 3}" (the cartesian product of {1,2,3} n times). (a) What is Sl? Give a brief explanation. (b) For 0 <k <n, let T be the set of all elements of S with exactly k occurrences of 3's. Determine |Tx I, and prove it using a bijection. In your solution, you need to define a set Ax that involves subsets and/or cartesian products with known cardinalities. Then clearly define your bijection...
10. -/3 POINTS LARTRIG10 3.4.044. Use vectors to find the interior angles of the triangle with the given vertices. (Round your answers to two decimal places.) (-2, -3), (2, 8), (9,2) • (smallest value) (largest value) -/1 POINTS LARTRIG10 3.4.049. Find u. v, where is the angle between u and v. || || = 90, || || = 250, 0 =
Complete the sketch of proof for Lemma 3.17: use theorems 3.16 and 2.5 f F is a finite dimensional separable extension of an infinite jheld Lemma 3.17. iEaa LenF. K(u) for some u ε . thern SKETCH OF PROOF. By Theorem 3.16 there is a finite dimensional Galois n field Fi of K that contains F. The Fundamental Theorem 2.5 implies that F, is finite and that the extension of K by F, has only finitely many intermediate AutA felds....
Example 13: Use Chebyshev's theorem with X = 68, n = 85 and S = 10 and do questions a- (a) Calculate the percentage (p) of data points that is within k =3 standard deviations of the mean. Substitute the given value of k into Chebyshev's formula and evaluate p • Write p as a percentage to one decimal place. p88.9% (b) Find the number of standard deviation (k) on either side of the mean that cuts off p=75% of...