2. Let A € M, (R) be given. (a) If AA = 0, prove that A = 0₁. Does this hold if A € M₂, (C)? (b) Suppose that A¹ A AA. Prove that A'A= A² if and only if A is symmetric. 2

Q: assume that Pr[A′]=0.55, Pr[B]=0.35 , and Pr[A′∩B]=0.2. (1) Pr[A∩B]=Pr[A∩B]=   (2)…

A: Solution : Given information; P(A')=0.55 P(A)=1-P(A')=0.45P(B)=0.35 and P(A'∩B)=0.2

Q: EXERCISE 5.7.2. Let S := {(-1)" nez}. Then Acc(S) = 10 11 (0) n. : no 013

A: This is a problem of accumulation points. Therefor it is a problem of point set topology.

Q: 1. Chapter 14 Review 43: Let g(u,v) = f(u³ - v3, v³ - u3). Prove that g д + u2 ди ди 2 = = 0.

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Q: (a, o) is not measurable.

A: to find that a,∞ is measurable or not  proceed as follows We know that every interval is measurable.…

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Q: The value(s) of x that satisfy In(x² + 1) – 2 In(x) = In(16) – In(8) are/is

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Q: Prove that if Xn X and Ync, then (Xn, Yn) → (X, c).

A: Please see the answer innext step

Q: |u| = |v| show that (u + v) • (v - u) = 0 is logically equivalent to condition u⚫u=v•v

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Q: -1 3 1 3 Is A stable? a) Let A = 4

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Q: Let a,b,c eZ. If a|b and a |c then a' |(b² =c² ). nsi2

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Q: Let T : R? → R² such that T (1,0) = (3, 1), and T (0, 1) = (-1,3). Find T (2, 5). T (2, 5)=( (enter…

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Q: 2. Let A € M₂ (R) be given. (a) If AA¹ = 0, prove that A = On. Does this hold if A € M₁ (C)? (b)…

A: This is a problem of Linear Algebra.

Q: Let A = -18 -15] 17 20 Find S, D, and S1 such that A = SDS-1 1 1 S- - BA] - PGI-1 D= = 1 2 0 -3

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Q: a(k) = eak coshw.k coshw, k ewok te-wok 2 eak coshw.k eak ewokteake-wok 2 X(Z) 3D글 z(enk e.k) + 글…

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Q: if zi = ayi for i=1,2,...,n, show that z=ay

A: Here we have given that if    then we have to show that   so here zi and yi are random variable…

Q: 1. Let A=(a)xn be symmetric and positive definite. Show that a) a>0, i=1,2,...,n. |a|< (aa)², i,…

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Q: 6. Let z E C. Let C be set of points z satisfying Im(z) 0 and Im Prove that C = {ZEC : |Z| = 1}.…

A: Given that  Im1+z+z21-z+z2=0 We know that if Imz=0 then Im1z=0 Therefore  Im1-z+z21+z+z2=0 Finding…

Q: 3. 43 Let A= Find nullity A") 1 8 --3 0 5

A: Nullity of a matrix : Let A be a matrix then  dimension of the nullspace of A  is called the nullity…

Q: (b) If WCU, prove that (Recall that U+W = (c) If U + W = U, prove that WCU. U + W = U. = {u+w|u€ U₁w…

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Q: s) Let ICS(P), 4,0, € S(P), and let I (p = 0). (a) Show that then riff I and (b) r, iff r, 0.

A: As per the question, we have to prove two statements related to the consistency of deductive…

Q: Prove that c?p?-(jx² + jy²+jz³) is the invariant quantity equal to c?po?.

A: Consider the provided question,   Prove that c2ρ2-jx2+jy2+jz2 is the invariant quantity equal to…

Q: Assume that =y+In(x), and n=x, then u xX а. O a. 1 Ob. 1 1 с. O c. x² d. 1 Unn x 1 O e. x² е. 2 +Unn…

A: ANSWER :

Q: 5. (8 points) Let A € M₂ (R) be given. (a) If det (A) = 1, prove that adj (adj(A)) = A. (b) If A is…

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Q: If A%3 2x3 -Зх +2:х € [О, 2) then in (R, T,), A %3D а. [-3, 1] b. [-1, 1] с. [1, 3] С. d. [0, 4] e.…

A: Open sets and Closed sets: The sets in topology τ are called the open sets of the topological space…

Q: Proof by contrapositive that for xy EZ, if xy is even then either x or y is even.

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Q: [2,5]U[3, 7]=[2,7]. (1,3)^(3,5) = 0.

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Q: Suppose A E C such that X(2+i, 3 – i) = (5- i, 7- 5i). Then (1) A = (1– i) (2) A = (2– i) (3) Does…

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Q: Suppose that the joint p.m.f. of Y and Y2 is defined by: 2x + y for x = 1, 2,3; y = 1,2,3 54 f(x, y)…

A: The probability mass function of Y1 and Y2 is,

Q: LET G= R\{-1}. Define * on G by a*b = a + ab+b. Find the solution the solution equation 3* x * 2=23…

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Q: 15. Suppose that a, b e Z and gcd(a, b) = 1. Prove that ged(a", b") = 1 for m, ne N. N and lot 8 frm…

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Q: Let ECR. Prove that m*(E)= 0 implies that E is Lebesgue measurable.

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Q: A := {x € (0,1)|x= ar, for some r € Q} re that sup(A) = 1.

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Q: k – 1 k+1 Let h : Z → Q × Q, given by h(k) : 2 2 Let G = {(m, m +1) | m E Z}. Show that G C h(Z).

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Q: Prove that if X₁X and Ync, then (Xn, Yn) (X, c).

A: For a sequence of random variables X1, X2, X3,... to converge to a random variable X, we must have…

Q: Q7: Prove the following: 1) Pr(A) = 1 – Pr(A) 2) Pr(S) = 1 3) Pr(A n B) + Pr(A n B) = Pr(A)

A: Since you have posted multiple sub-parts, we will solve the first three sub-parts for you. To get…

Q: 2. §11.3. Q52. Let w = √x² + 4y² - 2². Find (a). Əw/dx|(2,1,-1); (b). Ow/dyl (2,1,-1); (c). Ow/Oz…

A: The given function .The objective is to find the following,(a) (b) (c) .