2. Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication op- erations on u= (u1, U₂) and v (V₁, V₂): u+v= (u₁ + v₁ + 1, U₂ + V₂ + 1), ku= (ku₁, ku₂) www. (d) Show that Axiom 5 holds by producing an ordered pair 1 -u such that u + (-u) = 0 for u = (u1, U₂). (e) Find two vector space axioms that fail to hold.

Q: 1. Consider the set V of all polynomials of degree two or less, along with the following definitions…

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Q: Operation o additien for all (XuX), (XXanXa) E TR. Define an operation fScalar multiplication…

A: consider a set S on ℝ2 where define an operation of addition on ℝ2 by X1,Y1*X2,Y2=X1+X2+1,Y1+Y2+1…

Q: Let ū = (-1,2,4) and ū = (-3,5, 2). Compute the following: (a) ū+ū (b) .공 (c) ||해 (d) ũ - 7 (this is…

A: (a) Find u→+v→ as follows: u→+v→=-1,2,4+-3,5,2          =-1+-3,2+5,4+2          =-4,7,6

Q: Prove that Cov (x, y) = E (xy) – E (x) E (y).

A: Cov(x,y) is covariance of x and y. E(x) is expected value of x E(y) is expected value of y.

Q: 3. Consider the statements below. (I) For each b in Rnı the equation Ax = b has a solution. (II)…

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Q: Q.2:0Find the inverse the matrin A = -1 8 to (I A""). by converting Cii) P leas (A! 2) Comment on…

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Q: u = 3i – 2 j+4k v=i-3j-k 18) (Section 9.7) Find the cross product in space i j KI 3 -2 4 -3 -1|

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Q: Let Ai = {i, i + 1, i + 2, . . .} find ∩∞(infitity)i=1Ai and ∪∞(infinity)=1Ai Explain.

A: We have to find the intersection and Union of given set.

Q: use Gauss Fordanmetrecd to Solbe the fellowsing setafeg Qe linear 2X+ -4x2 +6X3=5 A +3X2-783=; + 5…

A: Introduction: A system of linear equations can be solved using Gauss-Jordan elimination method. This…

Q: [t +3 If A = -1 6. t- 3 -6 one of the following must be true? 1 t+ 4] A) The rank is 3 ift#-4, B)…

A: Rank of the Matrix, and it's properties

Q: 4 7 2 Let M-[and A-and-]-[3] B= 7 Express M as a linear combination of A and B and C 1) M=3A-2B+C 2)…

A: Given that the matrices  M=4779 , A=1111   , B=1234  and C=1145 .

Q: Prove that for operators A and B which commute with their commutator [Â, Â]= Ĉ, so that [Â, Ĉ]=[ B,…

A: Given that A⏞,B⏞ commute with their commutator C⏞. Now, ekA.Be-kA=B+kA,B+k22!A,A,B+k33!A,A,A,B+...…

Q: 2) Let R real numbers, define = x,Y1 +4x2y2 + 6x3Y3 for x = (x1,x2, X3) and y (y1,y2, Ya). Is inner…

A: Let V be a vector space over ℝ.An inner product <·,·> is a function V×V→ℝ with the following…

Q: Let V be the set of all ordered pairs of real numbers under the following operations of addition 1.…

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Q: if 3.^wa Proj, wiHen Proije, (-2w) = lal 3w

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Q: Given the vector (x <- c(5, 2, 7), what would x^2 return? O [1] 78 O [1] 196 O [1] 25 4 49 O [1] 10…

A: Given that, the vector is        ,we need to find .

Q: Prove in detail that ℚ(√3+√7) = ℚ(√3,√7).

A:

Q: Prov that Let A be a fuzzy ideal of R, anand by Lemma- be fuzzy singletons. Then (a.b) no A = anobu…

A: To prove that (a⋅b)h ⋅A=ah ⋅bh ⋅A given A is a fuzzy ideal of R and ah , bh are fuzzy singletons, we…

Q: 25) Which notation describes the translation of the sailboat B(70, 20) to the lighthouse B'(10, 80)…

A: Given: B70,20, B'10,80 For reaching from point B to point B', we have to use operation x,y⇒x-60,…

Q: Assume that n(U)=115n(U)=115, n(B′)=65n(B′)=65, n(A∩B′)=32n(A∩B′)=32, and n(A∩B)=19n(A∩B)=19 Find…

A: Now consider,

Q: What is the coefficient What is the coefficient of rx3x3 in (x₁ + x2 + x3 − x4 - x5)⁹? 1 13 of 2¹3…

A: To find the coefficient of x14 x32 x43 in (x1+x2+x3-x4-x5)9 we will use binomial theorem.  The…

Q: Let T be abounded linear operator on a Hilbert space H. Then T, =(T +T) is Select one: O none O (T'…

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Q: Let u = (-7, 1, 5) and 7 = (5, 3, 4). —

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Q: Let ---0--0 8 u= W= Compute 3w+6u = -131

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Q: Question: Show that the set Pn of all polynomials with real coefficients of degree n is a linear…

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Q: Let such that 1 0 1 -1 2 -6⠀⠀⠀-0-0 1 1 3 -1 b 000 0 Then there exist scalars b A True False = X1,…

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Q: Prove from fundamental definitions and principles, that (-00, 0] and (0, 4] are numerically…

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Q: 7

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Q: Need c) only use product rule 3. Differentiate. a) M(u) = (1– 4u²)(u+2) b) g(x)= (-x+ 3)(x- 10) c)…

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Q: Let a + 0. Assume that (5A – 3B) = I with | 1 2 3 A = 0 -4 -7 . Find ) 11 -1 the (2,3) entry of B.…

A: The given matrix is,A=1230-4-7011-1

Q: 9. Let V be the set of all ordered pairs of real numbers, and consider the following addition and…

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Q: Suppose J3 is derived from switching the rows of 13. -5 3 0 -8-1 7 -4 If AJ3 = 0 -2 7 -2 3 -5 -8 or…

A: NOTE: Refresh your page if you can't see any equations. . the 3×3 identity matrix is given by now…

Q: ao + 2a1 + a2 - azz²) = | da0 + 7a; + 5az 3ao + 5a, + 5a2, Let T(ao + a;7 - Determine ifT is…

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Q: Let H be a Hilbert space and A E B(H). Show that (ker A*)+. (a) Image A = (b) ker A = (Image A*)+.

A: