Prove that there exists an infinite quantity of solutions to x2=1in the quaternions. For example, in...
Prove that there exists an infinite quantity of solutions to x2=1in the quaternions.
For example, in H = { a + bi + cj +dk such that
a,b,c,d∈
} and i2
=j2 =k2 = i j k= −1
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Prove that there exists an infinite quantity of solutions to
x2=1in the quaternions.
For example, in...
1. The quaternions H are a set of "hyper-complex" numbers of the form p = a...
1. The quaternions H are a set of "hyper-complex" numbers of the form p = a + bi-cit dk. where a, b, c, d E R. Like the complex numbers, they can be added, conjugated by sending p ? p = a-bi-cj-dk, and the norm of a quaternion is given by lp-va2? 2 247. To multiply two quaternions, we use the algebra i2 = j2 = k2 =-1 ij=-ji = k jk=-kj = i ki=-ik = j a) Use the...
Create a Quaternion class that can handle quaternion algebras based on additions
In mathematics, the quaternions are a number system that extends the complex numbers.
Quaternions are generally represented in the form:
a + bi + cj + dk
where a, b, c, and d are real numbers, and i, j, and k are the fundamental quaternion units.
The product of 2 fundamental quaternion units has the following properties:Keep in mind the deadline is at the 1st of October 2022 so make it asap. The coding should also include meaningful comments as...
(TOPOLOGY) Prove the following using the defintion: Exercise 56. Let (M, d) be a metric space...
(TOPOLOGY) Prove the following using the defintion:
Exercise 56. Let (M, d) be a metric space and let k be a positive real number. We have shown that the function dk defined by dx(x, y) = kd(x,y) is a metric on M. Let Me denote M with metric d and let M denote M with metric dk. 1. Let f: Md+Mk be defined by f(x) = r. Show that f is continuous. 2. Let g: Mx + Md be defined...
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is inv...
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
Problem 15 (m* (I)) of an interval I is its length (e(I)). Prove that the Lebesgue...
Problem 15 (m* (I)) of an interval I is its length (e(I)). Prove that the Lebesgue outer measure (In): EC Ia (In) collection of open interva inf Proof Recall that the Lebesgue outer measure m' (I) n To prove that the Lebesgue outer measure is equivalent to the length of the interval, we will first 167 7.4. Measure Theory Problem Set 4: Outer Measure consider an unbounded interval I. Note that an unbounded interval cannot be covered by a fi-...
2 (b) Prove that + 3 cos(atx) O has at least two solutions with x €...
2 (b) Prove that + 3 cos(atx) O has at least two solutions with x € (-1,1]. [20 Marks] 1 + x2 (c) State the Rolle's Theorem. [5 Marks] (d) Prove that + 3 cos(1x) = 0 has excalty one solution in [0, 1]. 1 + x2 [20 Marks (Hint:Use proof by contradiction, by supposing more than one root. ]
Question ④ (20 marks) Consider a control system shown in Fig. I has an open loop TF-G(s) H (s)--( A-Prove that the gain margin-infinite db at infinite rad/sec. and the phase margin 62.1 degrees at...
Question ④ (20 marks) Consider a control system shown in Fig. I has an open loop TF-G(s) H (s)--( A-Prove that the gain margin-infinite db at infinite rad/sec. and the phase margin 62.1 degrees at 2.65 rad/sec.? B-Sketch the polar plot? 15 S(S+5) C- Sketch the Bode plot and show gain margin and phase margin? D-Sketch the Nichols plot? E-Write short MATLAB program to solve a, b, C and D? Best Wishes for all, Examiners
Question ④ (20 marks) Consider...
How many classes of solutions are there for each of the following congruences? (a) x2 -...
How many classes of solutions are there for each of the following congruences? (a) x2 - 1 = 0 mod (168) (b) x2 + 1 = 0 mod (70) (c) x2 + x + 1 = 0 mod (91) (d) x3 + 1 = 0 mod (140) Please note to show how you got the solutions as well as finding out how many classes of solutions there are for each congruence. Please explain every step so I can understand how...
Please show the solutions for all 4 parts! Problem 1 Let m E Z that is...
Please show the solutions for all 4 parts!
Problem 1 Let m E Z that is not the square of an integer (ie. mメ0, 1.4.9, ). Let α-Vm (so you have a失Q as mentioned above) (i) Prove the following:Qla aba: a,b Q is a subring of C, Za]a +ba: a, b E Z is a subring of Qla], and the fraction field of Z[a] is Q[a]. (3pts) (ii) Prove that Z[x]/(X2-m) Z[a] and Qx/(x2 mQ[a]. (3pts) i Let n be...
2. A Markov Chain with a finite number of states is said to be regular if...
2. A Markov Chain with a finite number of states is said to be regular if there exists a non negative integer n such that for any i, J E S, Fini > 0 for any n-มิ. (a) Prove that a regular Markov Chain is irreducible. (b) Prove that a regular Markov Chain is aperiodic (c) Prove that if a Markov Chain is irreducible and there exists k E S such that Pk0 then it is regular (d) Find an...
1. The quaternions H are a set of "hyper-complex" numbers of the form p = a + bi-cit dk. where a, b, c, d E R. Like the complex numbers, they can be added, conjugated by sending p ? p = a-bi-cj-dk, and the norm of a quaternion is given by lp-va2? 2 247. To multiply two quaternions, we use the algebra i2 = j2 = k2 =-1 ij=-ji = k jk=-kj = i ki=-ik = j a) Use the...
(TOPOLOGY) Prove the following using the defintion:
Exercise 56. Let (M, d) be a metric space and let k be a positive real number. We have shown that the function dk defined by dx(x, y) = kd(x,y) is a metric on M. Let Me denote M with metric d and let M denote M with metric dk. 1. Let f: Md+Mk be defined by f(x) = r. Show that f is continuous. 2. Let g: Mx + Md be defined...
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
Problem 15 (m* (I)) of an interval I is its length (e(I)). Prove that the Lebesgue outer measure (In): EC Ia (In) collection of open interva inf Proof Recall that the Lebesgue outer measure m' (I) n To prove that the Lebesgue outer measure is equivalent to the length of the interval, we will first 167 7.4. Measure Theory Problem Set 4: Outer Measure consider an unbounded interval I. Note that an unbounded interval cannot be covered by a fi-...
2 (b) Prove that + 3 cos(atx) O has at least two solutions with x € (-1,1]. [20 Marks] 1 + x2 (c) State the Rolle's Theorem. [5 Marks] (d) Prove that + 3 cos(1x) = 0 has excalty one solution in [0, 1]. 1 + x2 [20 Marks (Hint:Use proof by contradiction, by supposing more than one root. ]
Question ④ (20 marks) Consider a control system shown in Fig. I has an open loop TF-G(s) H (s)--( A-Prove that the gain margin-infinite db at infinite rad/sec. and the phase margin 62.1 degrees at 2.65 rad/sec.? B-Sketch the polar plot? 15 S(S+5) C- Sketch the Bode plot and show gain margin and phase margin? D-Sketch the Nichols plot? E-Write short MATLAB program to solve a, b, C and D? Best Wishes for all, Examiners
Question ④ (20 marks) Consider...
Please show the solutions for all 4 parts!
Problem 1 Let m E Z that is not the square of an integer (ie. mメ0, 1.4.9, ). Let α-Vm (so you have a失Q as mentioned above) (i) Prove the following:Qla aba: a,b Q is a subring of C, Za]a +ba: a, b E Z is a subring of Qla], and the fraction field of Z[a] is Q[a]. (3pts) (ii) Prove that Z[x]/(X2-m) Z[a] and Qx/(x2 mQ[a]. (3pts) i Let n be...
2. A Markov Chain with a finite number of states is said to be regular if there exists a non negative integer n such that for any i, J E S, Fini > 0 for any n-มิ. (a) Prove that a regular Markov Chain is irreducible. (b) Prove that a regular Markov Chain is aperiodic (c) Prove that if a Markov Chain is irreducible and there exists k E S such that Pk0 then it is regular (d) Find an...
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