Exercise 6.65. Prove that if E₁ CR and E₂ C Rm are convex, then E = E₁ × E2 is a convex subset of Rn+m. Hence the Cartesian product of n intervals in R is a convex subset of Rn.

Q: b. Determine the value(s) of b such that the set S = is linearly independent in M

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Q: ercise 6.67. Let E be a subset of R having at least two points. Pro- owing are equivalent:

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Q: how that the set S={(1,2,1),(3,1,5),(3,-4,7)}  is linearly dependent where, S⊆V\index{3}(R).

A: Our Claim is to show that Set S is linear dependent in V\index{3}(R).Since we know that if u,v and r…

Q: 19. Prove that if {A₁, A2,. Ak} is a linearly independent subset of Mnxn (F), then {(A, A₂,..., At}…

A: 19. Prove that if  is a linearly independent subset of , then  is also linearly independent.

Q: › affine space (E‚È‚+), let (ai)ie is linearly independent for some i

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Q: Consider the net revenue function given by S(p, u, v) = R(p, v) – E(p, u) in our usual notation.…

A: Use this net revenue function to determine the correct option.

Q: The column space of A is the set of all vectors that are linear combinations of the columns of A. 3…

A: Given matrix A=24-21-2-57337-86  and  u=3-2-10  and  v=3-13 Remarks If AX=0then x in Null(A)

Q: 2. Let x be an element of a convex set S. Assume that x1 = x + €ip E S and x2 = x – €2P E S, where p…

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Q: Exercise. Consider the set R. For every x = (x1, x2) E R? and y = (y1, y2) E R², write Pi(x, y) =| x…

A: (a) Here, ρ1x, y=x1-y1+x1-y2   (1) show that ρ1x, y≥0 : We know that, x≥0,  ∀x Therefore, x1-y1≥0,…

Q: II. Show that if preferences 2° € X*. strictly convex, then A'(xº) is strictly convex for each are

A: Use the preference criteria to prove the following. Show that in each case, they are convex.

Q: P3.6.8 Problem: Consider the set S = ZV2] as defined in Exercise 3.6.6, a set of real numbers.…

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Q: 1. Show that ||x|| , ||x||||x||₂ are vector norms. 2. Show that if x and y are two vectors, then…

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Q: 1. Let T: VW be a linear map. Let (v₁,...,Un} be a list of n vectors in V. Prove that if the vectors…

A: Let  T:V→W be  a  linear map. Let v1,v2,...,vn  be  a  list of  n vectors in  V.Suppose  that the…

Q: 1. Complete each statement for the translation T:(x, y) → (x + 3, y - 1). a. T glides points units…

A: T:(x,y)→(x+3 , y-1)

Q: Prove that the Frobenius norm is consistent. That is, for A, B € Rxn, show that ||AB|| ≤ ||A||||B||F

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Q: Let M = {(0,0),(0, 1)} be subset of R, then M is linearly independent %3D O True O False

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Q: Let A = [1 2 3 5 4 6 7 8 Find the dimensions of Nul A, Col A and Row A.

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Q: (3) S° is the largest open set in M that contained in S.

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Q: EXERCISE 8.3 Eor.1 and 2, let u=(5,5) and v=(3,-2) 1.) Verify that ||–4u ||=||4 || |||| 2.) Verify…

A: The given vectors u=(5,5) and v=(3,-2)

Q: ) Let z be a particular solution to a linear equation system Az = b. Let {u, u2, uz} %3D be a…

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Q: Assume that AA and BB are disjoint subsets of UU, and that n(U)=55n(U)=55, n(A)=23n(A)=23, and…

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Q: 1. Let X={a,b,c} and Y={1,2} then X×Y ={(a,1),(a,2).(b,1).(b,2),(c,1).(c,2)} - Consider U =…

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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: Let An denote the number of lattice paths from (0,0) to (n, n) that do not pass above the diagonal y…

A: As per the norms, we will be answering the first question. If you need an answer to others, then…

Q: Determine whether the set [p., pa, p3] is linearly Independent in Pa, where pi= 3+x+3x²+4x3 Pa = 4+…

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Q: 1. Decide if the following statement is True or False: %3D Let A1, A2, ..., Ak E Mnxn(R) be such…

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Q: 4. Let r E R" be such that r = (E1,... , En). Show that the norms ...+ En and = V are equivalent so…

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Q: The population of a community is known to increase at a rate proportional to the number of people…

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Q: Set X is convex if

A: Given that a set X is convex if  . . . .

Q: If V = W1 •…. Wk, then there exist k linear operators E1, … , Ek On V such that %3D (i) each E; is a…

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Q: 3. Find the value(s) of a for which ở = (-1, 2a – 1,3) and w = (2,2a + 3,–6) are parallel.

A: Two vectors  are parallel  is their cross product  is zero. So calculate  cross product  in the…

Q: Let V be a 5 dimensional space, and V: V → V linear. Can we have NullT = RangeT? And if the…

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Q: Let M = {(1,–1), (2, 2)} be subset of R², then M is neither independent nor dependent O linearly…

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Q: Suppose that W, X, Y, and Z are points with X between W and Y and Y between W and Z. Prove, using…

A: It is given that W, X, Y, and Z are four points such that X between W and Y and Y between W and Z.…

Q: In R4, let S = span{(1, 1, 1, 1)T ,(1, 2, 1, 3)T }. Find S⊥.

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Q: Which of the following sets is linearly independent? * 21 [2 the above set None of these ([o o1 [1…

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Q: Which of the following statements are true for the set S= {v,, ..., V} in R"? 1: {v2 - v1, ..., Vk -…

A: We are given the set S=v1, v2, ..., vk in Rn. We know that,     The affine hull aff(S) of S is the…

Q: Consider R? and R3 with the Euclidean (c) norms. Compute || T || for T : R2 →R3 defined by T (x1,…

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Q: Let AE Rmxn and suppose that the rank of A is m where m is less than or equal to n and be Rm.…

A: Given: We are given a matrix A of size m×n with rank m where m≤n, a vector b of length m×1, and a…