Let A be an wmxn matcix Prove that %3D

Q: Exercise 2. Show that if B + Ø then dom(A × B) = A. %3D

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Q: Show that Q(i) is a subfield of c

A: Subfield of a field

Q: Let A be an mxn Aqtrix Prove that = max |Ax/ -max E l9;

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Q: a, Find at2

A: Given :…

Q: Prove that fVf · dr = 0.

A:

Q: Let A E Mnxn(R), show that it is possible to express A as the product, A = PQ where P,Q € Mnxn(R),…

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Q: Let A Find grad(div( A ) %3D

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Q: 2. Show that cÃ=cA.

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Q: The characteristic of Z6 x Z, is 18.

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Q: Let a e Z. Prove that if 6 | a and 10 | a, then 15 | a.

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Q: Let X ∼ Bin(n, p), and let Y = n − X. Show that Y ∼ Bin(n, 1 − p).

A: Here X follows Binomial with parameters n and p. Here, Y=n-X. Then Y takes the values n-0, n-1,…

Q: Use the result of Exercise 41 to show thatf1−α,ν1,ν2 = 1fα,ν2,ν1

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Q: Prove that if AT A = A, then A is symmetric and A = A².

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Q: Prove that A CB P(A) C P(B).

A: We have to prove that

Q: Notice that h is definec this.)

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Q: Find the characteristic of Zm, × Zm, × · · × Zmr where m;'s are positive integers.

A: The ring ℤm1×ℤm2×...×ℤmk is a ring with unity 1,1,1,...k times Now, to find characteristic of…

Q: Prove that if A, B ϵ Mnxn(F) are similar, then det{A) = det(B).

A: Given:  A, B ∈ MnxnF are similar To prove:  det(A)=det(B).

Q: @S. Let fGJ = 12 Find fla)=

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Q: Find the Re and Im parts of exp(z?)

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Q: Exercise 2. Prove that if A = B and C z D, then A × C = B x D.

A: Given data: A≈B......(1)C≈D......(2) The objective is to prove A×C=B×D

Q: Use the proposition on the expected product to show that when X and Y are independent, Cov(X, Y) =…

A: If X and Y be two random variables, then the correlation coefficient between X and Y is given by,…

Q: = Prove that E = ehd.

A: We know that Ef(x)=f(x+h)

Q: If A is similar to B, prove that AT is similar to BT.

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Q: Let a in F_p, a is non zero then show

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Q: Let a, be such that asb. Prove that tx Eqt ax cbx

A: Concept: We have a,b∈Q means that a, and b belongs to the rational numbers and a<b means that a…

Q: Prove that OfVg.dr = ] (Vfx Vg) · N ds.

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Q: Prove that det(At) = det(A) for any A ϵ M2x2(F).

A: Suppose we are given a square matrix A with four-element: a,b,c,d A=abcdAt=acbd

Q: Prove: Provided it exists, (A-1)¯ = A.

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: If z, w E C, prove that z+w=z+w.

A: This follows from simple substitution. The detailed solution is presented in step 2. As requested i…

Q: z has a power (z) is

A:

Q: Consider the following theorem.

A: Given, Where, And We have evaluate the integral,

Q: Let Evaluate z dv.

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Q: Let a e Z. Prove that if 7 | 2a, then 7 | a.

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Q: 2) Let a,bE Q such that a<6. Prove that ax <bx.

A: The statements can be proved by analyzing the results from the given statements. The other methods…

Q: Let z, w e C. Clearly prove that z+w = z + w.

A: Since you have asked multiple question , we will solve the first question for you. If you want any…

Q: Prove Theorem  A + (B + C ) = (A + B ) + C .

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Q: Prove That A. CBxc) = B:(C xA)

A: As per our guidelines we only solve one question and rest of the question will be reposted.

Q: Suppose A · B = 0. Show that this implies that C(B) is contained in Null(A).

A: Suppose A.B = 0 then we have to show that  Column space of B contained in the null space of A  i.e…

Q: Evaluate (x+ V1- x²) dx modd ear od wod woml bre hJo 1osr esirmol lesgsta odinia by interpreting it…

A: The given integration is: ∫01x+1-x2dx To find the value of the integration.

Q: . Let X be a r.V. such that E(X") is finite for all n = 1, 2,... Use the expansion

A: Introduction: Suppose X is a random variable. Then, the moment generating function (mgf) of X is…

Q: Let A be uncountable and B be countable. Prove that A\B is uncountable.

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Q: Show that a) A − ∅ = A. b) ∅ − A = ∅.

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Q: In a

A: We have to find given indefinite integral.

Q: Which expression is an expanded form of In(t)?

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Q: (a) Show that for x > 0, 1 5a' < In(1+ x) < |

A: As per guidelines, we will solve only first part. Kindly repost other part separately.

Q: (a) Prove that if f e C (R²) then a (af af %3D

A: This is a problem of the calculus of several variables, higher dimensional calculus.

Q: Let x(v, w) = } In(v² + w²). Show that = 0

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Q: Prove that (aA + bB)t = aAt + bBt for any A , B E Mmxn(F) and any a,bϵ F.

A: Consider, aA+bBt=aAt+bBt    ∵A+Bt=At+Bt=aAt+bBt     ∵kAt=kAt for any scalar k.=aAt+bBt

Q: Prove that S[x] = {a+bx where a,b in R} is subring ℝ[x].

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Compound Probability

Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent. 

Tree diagram

Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described. 

Conditional Probability

By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.