4. Suppose d € Z with d > 1. This ensures that √-d & Q. Prove that Z[√-d]× = {±1}. (Hint: use the norm.)

Q: 4. If [æf (x) + 2] dæ = x³ – a² then find the function f (x). 3 A) 2 3 B) 2+ C) 3- 2 E) 2– D) 3+ --…

A: The given equation is x∫x fx+2dx=x3-x2.

Q: Suppose g [a, b] → R is a continuous function that takes only irrational values. Prove : that 9 is…

A: Use the Intermediate Value Theorem (IVT) and method of contradicts.

Q: are orthogonal on einmx/L, n = 0, ±1, ±2,... [-L, L] with norm equal to √2L.

A:

Q: 4. Let F = {s+t√2 s,te Q}. Show that F is a subfield of R. You may assume that √2&Q.

A: The objective of the question is to prove that the set F, defined as F = {s + t * sqrt(2) | s, t…

Q: Exercise 6. Let f(x) = 2x-5logx. Then f(x) is а. О(х) b. O(x²) с. О(Зx?) d. All of these

A:

Q: 5) Find the divergence of (√x² + y² + )(i + y + zk)

A:

Q: Exercise 2. Prove that dim(null T") = dim(null T) + dim W – dim V and dim(range T") = dim(range T),…

A: We prove dim null T*=dim null T + dim W-dim V and dim range T *=dim range T for every T L(V·W).

Q: Show that A = (aj) = Cmxn. Then || A||2 ≤ || A||F ≤ √ Rank(A)|| A||2 V

A: Matrix A=(aij)∈Cm×n: This notation indicates that A is a matrix with complex entries, where m and n…

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

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Q: 6. Let F = (-6, 5)U{-10, 10} U [7, 11). Find max F, min F, sup F, and inf F if they exist.

A:

Q: 2. Suppose Y is G-measurable. Prove that Var(Y/G) = 0. %3D

A: Given that Y is G-measurable.

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

A:

Q: In R, the model mod <- Im(z ~ x + y) will contain: O the results from regressing the dependent…

A: Introduction: It is required to identify the correct option.

Q: Evaluate the bound of the error if the Trapezoidal rule applied to S (x)dx . Given that 5 k is…

A: Apply formula of trapezoidal error

Q: Solve using the method of undetermined coefficients: y" + 3y" – 10y' = 2 + xe* %3D

A: Given: y''' + 3y'' - 10y' = 2 + xex

Q: -Prove that 2x² - 8x = 22 has at least one soution on the interval IVT. [5,10] by using

A:

Q: lineari 2ation L(27 of f(z) that L (2.99) grves approximation for f(2,99). Find the at the integer…

A: Consider the provided question,

Q: Which of the following function is NOT Q, (n)? А. п —333п? nlgn С. п-1gn D. 219 n E. None Е.

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Q: t K = F(u), u algebraic over F and degree of u odd. Show that K = F(u²).

A:

Q: Let X=IR". Let ||. =max {|a 1\, |a 2 ..a n}, x=(a1, a2 . a n) eIR". (αν α, Show that || · ||. is a…

A:

Q: Exercise 4. Suppose TE L(V) is normal. (a) Prove that range T = range T*. b) Prove that null T* =…

A: Given that T∈ℒV is normal.  To prove that range T=range T* Since T is normal, for all v∈V, Tv=T*v…

Q: 3. Let h: Z→ Z be the function h(r) = = x mod 4. Find h-¹(0), h-¹({0, 1, 2, 3}), and h-¹(4). Explain…

A: In this question we can easily observe that that output of the function is the remainder when any…

Q: Fad he averge yalue of the funchibon f fix) over the loterual [o,7] 3e- %3D

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Q: Let f(x) = VT, 9(x) = ,. , andh(x) = x². (h o ((h o go f) – f)) (4) = %3D 5-x Select one: a. 4. 81…

A: let f(x) = x , g (x) = 45-x and h (x) = x2f (4) = 4= 2g (f(4) = g2= 45-2= 43h (g(f(4)) = h 43=…

Q: 2. Find the (infinity norm) condition number of 2.01 1 (b) A = 3 6.

A:

Q: 21. 1 0 -B 0 Let V₁ = 1 0 | 2 -1 0 1 {V1, V2, V3} span R4? Why or why not? V3 = " . Does

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Q: 5. Determine the vclue of c such that the function is continuous on (-o0,00 + flx) = 2x, x SC

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Q: Find norm of the function in In Space L1 [o13 O a. 15/81 O b. 15/324 O c. 17/81 O d. 17/324 O e.…

A:

Q: GeoGebrä Task 1: This rectangle has been decomposed along its diagonal. Recompose the two pieces to…

A: A rectangle is a closed quadrilateral with four sides and four angles. All the four angles of the…

Q: einπx/L n = 0, ±1, 2, ... re orthogonal on [-L, L] with norm equal to √2L.

A:

Q: 5. Calculate the divergence of F = (7x², 9y, -52).

A:

Q: 95. Prove that ||v - w2 = ||v|2 + ||w|2 – 2v • w. 97. Prove that ||v + w|[² + |v – w|? = 2(||v||² +…

A: Note:- We’ll answer the 95th question since we answer only one question at a time. Please submit a…

Q: (b) Assume C and D are chosen at random from {1,2., n}. Let pn be the probability that C + D is a…

A: Given: C and D are chosen at random from {1, . . . , n}. And pnis the probability that C + D is a…

Q: 32. Show that, in a normed space, x x implies -(x1 ++Xn) → x.

A:

Q: (b) Define the operator T: → as: (Tx) = for every x = () e t. That is, Tx = (61. for every x (f,E2.…

A: