24. TIO ieing generalization of the Linear Combination Lemma. Let mi, m2 If d x, for i 1,2,. . ., 7, then d m%2mx my x1 , 2) .. x, EZ is a natural number. 25. Let a and b be natural numbers. Suppose that Prove that either ab 1,or a b 2. 3.1 Basic Properties

Q: 1. Let x, y, z E R, we R+. Also let a, b, ceZ and d ez*. For each of the following determine is…

A:

Q: 9 t Suppose fe R[a, b] and m < f(x) < M, for all æ € [a, b). Prove: m(b – a) < f(x) dar < M(b – a).

A: Given: f∈Ra,b and m≤fx≤M,for all x∈a,b. To prove: mb-a≤∫abfxdx≤Mb-a

Q: 4. Let aj = 9. and b = 63 , a2 = 45 (a) Write b as a linear combination of a1 and a2. b = aj + a2.…

A:

Q: 11 6 linear combinations of x, y, and z, or show that it cannot be done. Let x = 120 and z = TTTT If…

A:

Q: 2 20 Let A = Which of the following is a consequence of the Cayley -Hamilton Theorem? 2 10 L-10 0]…

A:

Q: 2. Let A, B, and C be as in Exercise 1, let D = {3,2}, and let E: Determine which of the following…

A: "Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: (1) Show that if E(X), E(Y) and E(XY) are well defined then Cov(X, Y) = E{(X – E(X))(Y – E(Y))}. The…

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

Q: 3. Consider the follow Twitter by 16 high s 0.8 36.7 3.7 14.9 a) Find all measur b) Find all measur…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: let f:(a,b)→R be de fined by X€QN(a,b) x€Q' n(a,b) -2 f(x)=' 3

A: This is a problem of Differential Calculus.

Q: 4. (а) Let a, b, c E Z. Prove that, if a | b, then ac | bc, but that the converse may be false. (b)…

A:

Q: If T : R6 [a] → M2x2 is onto then Nullity(T) = 3. O True O False

A:

Q: Lt A 28 be anbitrary closed subsets 9 (X,d) 3f A°=4 e B'= ¢ the give Such That AUB)"=X. erample an

A:

Q: 5. Let z1 = a1 + b;i and z2 = az + bzi. (a) Find Im[z,Zz + (Z)? – 2,7 ]. (b) Prove that (z)² = z,² .

A: Rational Numbers: All the numbers which can be written in the form of pq. Here, p and q are integers…

Q: 2. If A C B, then m*(A) < m*(B). 3. If A = Q' n (0,1), then A is measurable. %3D

A:

Q: (3. Suppose T is a linear tronstormation from V to w and that {š X is a LI. söt of ve ctors in V.…

A:

Q: If T: R"„8* is a Linear transfor matiom and m (x)= (2² +2)Cx+ 3. Find all the ratinal canomical ferm…

A: Companion matrix:  Let f(x)=xn+an-1xn-1+………+a0 be a polynomial then the matrix form is given by…

Q: 1.1 Let z, w e C. Show that |z + w? + |2 - w? = 2(|:|2 + |w?).

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

Q: 20. Let (). ()- u = v = If æu + yv = w. (æ, y E R), then a - y = (A) 2/3 (В) —3/2 (C) 3/2 (D) –1/2…

A: To find the value of x-y from the given vectors.

Q: (13) Let a, b, c, d be constants. If all of the followving covariances are well defined then show…

A: Let a, b, c, da, b, c, d be constants. Cov(aX + bY, CZ+dW) = ac Cov(X, Z) + ad Cov(X, W)+ bc…

Q: Let a ER and set fa: R→R: xx+1|-2|2x|+ a. S(a)= {r ER: fa(x)=0}. (So, S(a) is a set). P= {a € R:…

A: Given information fa : ℝ→ℝ  : x→x+1 -22-x +a Sa =  x∈ℝ : fax =0  P =  a∈R : Sa = ϕ Qa =  x∈ℝ : fax…

Q: Prove that, if (a%m)=(b%m) then m| (a-b)

A:

Q: For all a,b€ R, if|a – b|< min(a,b), then prove that max(a,b)< 2min(a,b). く

A:

Q: It gER[a,b] and if h(x)=q(x) except for a finite number of points in [ab],then prove that he R[ab]…

A: The continuous set of [a,b] should be divided into finite sets first.

Q: 1. Let D= [-2,3) U {-3, T}. Determine the following: (a) max D, (b) inf D, (c) interior ED (a) b

A:

Q: 2. Prove the following statement directly. V a, b, c e Z, if a | b and a c, then a | (7b + 4c)

A:

Q: 8on Dehe Left e and Ryht hand 8onDehe Leff hand Deivatie Rght handd Deivatai Explan with evamle =…

A: First we have to write definition of Right Hand derivative as well as left hand derivative. a)

Q: Which of the following statements is false A. {2,3} CA implies that z E A and 3E A- B. A – B 2 {3}…

A: Sets true or false

Q: A ber coatais 6 whine, 4 bMar and 5 red balk. 4 balh ar chea at randem Let X denote the ofblae bub…

A: We have to find given probability.

Q: 5. In each of the following cases, either express p as a linear combination of u, v, w or explain…

A:

Q: 27. Let a, b eR 3 a> 0 and b > 0, A = (0, a) and B = (0,b). If D = {xy : x € A and y e B} %3D then…

A: Option (C) is correct.

Q: The greatest common divisor c, of a and b, denoted as c = gcd(a,b), is the largest number that…

A: Here is the solution

Q: Let X be an RV. If Var(−2X+1)=5 and E(X)=1.5, then E[(X−2)2] is  Select one: a. 1.5 b. 1.25 c. 4.5…

A: Expectation of a Random Variable: The expectation of a random variable X, denoted by E(X), is…

Q: Complete the proof that ATUW ARXV. W R Statement Reason WX UV Given 2. RX TU Given ST RS Given 4 SW…

A: 5. RV = RS + SV [Additive property of Length] 6. TW = ST + SW [Additive property of Length]

Q: If w=f(p, q, r), where p-z2-y², q=y²-22, and r = 22-2², show that 1 Ow x 8x + 1 8w у ду + 18w z Əz…

A: Given w=f(p,q,r)and p=x2-y2,   q=y2-z2 and r=z2-x2

Q: 7. Let w = 8x? + 6 y² +2z° ;x = r²; y =t*; z =r² Find A) -321 + 481- + 201º В) -321 +481' + 201° - =…

A:

Q: Let S = {n + m√2 : n, m ∈ Z}  To prove that the sum of any two elements of S is an element of S.…

A: Given: S=n+m√2:n,m∈Z  (a) Let a+b2∈S for all a,b∈Z and let c+d2∈S for all c,d∈Z Add both these…

Q: 44. (a) Let a, b, ceZ with (a, b) = 1 and a | bc. Prove that a c. (b) Provide a counterexample to…

A:

Q: (a) Let E = {q € Q : V2 < q< 4}. Find sup E, inf E, E, and min E if they max exist.

A: The given problem is to find the Sup E , Inf E , Max E , Min E for the given set E. Given set E ={ q…

Q: -2 Let m be a real number and M = {(" ,). )}: 9- If M is a linearly dependent set of M2,2 then m=1…

A:

Q: Let P= {p:p=6x-1, for some x EZ} and Q= {q:q=6y+5, for some y €Z}. Prove that P= Q

A: Answer

Q: w that the set S = 3 6. 4 5 ms all of R³ or find an element of R³ not belonging to span(S).

A:

Q: 2. Let A, B, and C be as in Exercise 1, let D = {3,2}, and let E = {2,3, 2}. Determine which of the…

A: See solution below. Properties of sets have been used.

Q: 6. Given M = {(x, y), 1 < x² + y² < 4} C R². (a) Determine dM and ÔM. (b) Is M connected? Pathwise…

A: Given that M=x,y,1≤x2+y2&lt;4⊂R2 (a) The objective is to find dM and ∂M dM=x,y|x2+y2=1 and,…

Q: 4.4. Suppose that u = (1,3) and v = (3, 1). %3D (a) Sketch the points ļu + žv, ļu + ļv, and u + tv.…

A: Hello. Since your question has multiple sub-parts, we will solve the first three sub-parts for you.…

Q: 4. 3 Determine if bis a linear Let a, = and b = a. combination of a,. a, and a,.

A: Here, we need to verify whether b is a linear combination of a1, a2, and a3. Let…

Q: [:! -8 -2 -9" 27 1 Determine if w is in Col A. Is w in Nul A Let A = 4 8 Jand w = 4 64

A:

Q: Which of the following is equivalent to the statement "If any book that any member has checked out…

A: The statement given is as follows: "If any book that any member has checked out is overdue, that…

Q: (1) -) Let o, t e S6 such that 1 2 3 45 6 1 T = 3 4 and 6. 4 3 5 1 2 1 6 5 4 Determine the…

A:

Q: Let D = {2,5}, E = {1,4, 7, 8}, and F = {2,6, 10}. Determine whether the following statement is true…

A: Given D=2,5, E=1,4,7,8 and F=2,6,10. Consider the statement "∀x∈D, ∃y∈E such that ∀z∈F, x is odd…

Q: [-8] Let v1 = 6 ,v2 =| 4 ,V3 = 8, and p =| 1 Determine if p is in Col A , where A= [v, v2 vz]. (a)…

A: