Let V = {(a1, a2): a₁, a2 € R}. Define addition of elements of V coor- dinatewise, and for (a₁, 02) in V and c € R, define c(a₁, a2) = (0,0) if c = 0 [(ca₁, a²2) if c ‡ 0.

Q: Determine integers j, k, u and v so that the set S = {(j+kn, u + vn) : n E Z} is equal to the set T=…

A: Equation  157x+107y = 1 is a linear Diophantine equation (ax + by = c). Here , a = 157 b = 107 and c…

Q: Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (х, у)…

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Q: Show that the function below is not an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2,…

A: We have to show that the function below is not an inner product on R3, where u = (u1, u2, u3) and v…

Q: Define A: C → C° by A(z,, Z2, Z3) = (a,z1, azz2, lzz), «; e C. When is A a positive operator ?

A: A is a positive operator if all the eigen values are non negative Therefore, T(1, 0, 0) = (α1, 0,…

Q: Let u = Al 2 [R and v= find a and b such that au + bv [:] 3

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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: 4. (a) Let A € R² be open and let f: AR be C². Let (a, b) A and suppose the rectangle R= [a, a +h] x…

A: (a) Since f, using Mean value theroem w.r.t. x for two variables there exists a pR such that f(a +…

Q: ppose S = &v₁₁ ore that T = nearly independ

A: Answer given below.

Q: 1. Answer all parts. (a) Consider the operators A and B. Show that (i) (ii) [A, B]=-[A, B] [Am, A"]=…

A: Operators :A and B

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: Let A ∈ M (3,R) be invertible and C, D ∈ M (3,2,R) . Solving B from the equality A(B+Ct)t = D, what…

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Q: assume that Pr[A]=0.25Pr[A]=0.25, Pr[B]=0.35Pr[B]=0.35, and Pr[A∩B]=0.1Pr[A∩B]=0.1. 1)Pr[A|B] =…

A: Given: P(A) =0.25 P(B)=0.35 P(A∩B) =0.1

Q: Let Z[V2] = { a+bv2 la,b e Z}. Define addition and multiplication on Z[V2] (a + bv 2) + (c + d/2) =…

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Q: 8. Let V={(x,y)x,y € R}. Define addition on V as follows: (x»x)+(x,y)=(x +2x, 1,y+Y,+2) Find the…

A: To find zero vector of V

Q: Let U = 3i−j−4k and V = −2i+4j−3k. Show that U·V is commutative but U×V is not.

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Q: Let X and Y have the following joint probability density function: fx,y(x,y) = 1 / (x² y²), for x>1,…

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Q: Let Ps have the inner product given by evaluation at-2,-1, 0,1, and 2. Let p(t) = t2- 2t and q(t) =…

A: Let P3 have the inner product given by evaluation at -2, -1, 0, 1 and 2 . Let p(t) = t2-2t  and q(t)…

Q: Q.4.(a). Show that the following functions is not an inner product on R3, where u = (u, U2, U3) and…

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Q: a) Show that under the transformation: k = ab the equally spaced z-values, z : 10, 7o + h, 7o + 2h,…

A: The given transformation is, k=x-(x0-ah)h. When the value of x is x0 the value of k is as follows.…

Q: If |ū| = 4, |v|=5, |ū-v=6, determine |ū+ v.

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Q: Given u = (0,1, -1,2) and v = (1,1,3,0) find ||u + v||

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Q: Consider the set S = {p, q, r}, where p=1+x, q= x-x² and r= 1+ 2x². Show that the set S is linearly…

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Q: Which of the following must be true? ) Let V = {(x, y) : x, yeR}For any (x1,Y1), (x2, Y2)eV and c…

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Q: Find the values of w, x, y, z and Dx, using Cramer's Rule x + y + z + w = 13 2x + 3y - w = -1 -3x…

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Q: Suppose AE R, v#0, and (A- A)v#0. Use the fact that A is symmetric show that it is impossible to…

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Q: Let T:P2(R) → P2(R) T(a + bx + cz?) = 3a – 16 + 6c + ( – 1la + 4b – 22c)z + (– 6a + 36 – 1lc)z? Let…

A: Let us us consider the mapping T:P2ℝ→P2ℝ be defined by Ta+bx+cx2=3a-b+6c+-11a+4b-22cx+-6a+3b-11cx2…

Q: a) Define AAB = (A − B) U (B − A), where A, B are any two sets. Show that (ΑΔΒ)ΔC = ΑΔ(ΒΔC).

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Q: Show that u and v are orthogonal if and only if ||u+v|| 2 = ||u|| 2 + ||v|| 2

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Q: The div A at p(1,1,2) for A is .... Å = x²y i + 2 x z j + 4 - z² y k - 3 ( 3 зі зј о

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Q: Let W = {(x, y) E R² : x, y > 0}. Define addition + of any two elements (x, y), (w, z) E W by (x, y)…

A: Let u=(x,y) and v=w,z be any two elements of W, where w,x,y,z>0. Use the definition of addition…

Q: Let $(u, v) = (u², v). Find the following images under (a) The u-axis O the positive x-axis…

A: Given, Part a)  i) To find the image of u-axis under Since the v coordinate of points on u axis is…

Q: = Let p(x) = x³ x² + 2x + 3, q(x) = 3x³ + x² − x − 1, r(x) = x³ + 2x + 2, and s(x) = 7x³ + ax² +5.…

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Q: Let V be the set of all pairs (x, y) of real numbers together with the following operations: (x₁,…

A: V be the set of all pairs (x,y) of real numbers  together with the following operations ;…

Q: Show that the operators T,··, T4 from R² into R² defined by ... (§i, 52)→ (Š1,0) (§1, É2)→ (0, E2)…

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Q: Let →a=⟨7,−5⟩ and b→=⟨1,3⟩. Compute the dot product. →a X →b=

A: Given a→ = 7,-5 and b→ = 1, 3. We have to compute the dot product a→·b→.

Q: Let f(x, y) = In(x²y), where a = s² + t² and y = 2s + 3t. %3D Which of the following is a correct…

A: Given,           f(x,y) = lnx2y3,  where x = s2 + t2 and y = 2s + 3t

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: Evaluate f (x - y)(y - z) dV for the box B3 with dimension: [4, 9] × [0, 1] × [0, 1]. (Use symbolic…

A: We have to evaluate the integration .