3. Exercise $6.5 #34. Prove that a positive integer is expressible as the difference of two squares of integers if and only if it is not of the form 4n +2, n € Z.

Q: Prove that (-1+ iE)" + (-1- i5" has either the velue 2n+! or the value -2" f ロ ww. š any integer…

A: The cube root of unity is written as:

Q: Prove that Ją (2) = =- J¼ (2) – J_4 (=)

A:

Q: (a) For all a, b e Z prove that a³b² – ab* is even.

A: Properties of algebra Addition of two odd number is even  Addition of two even number is even…

Q: Let a and b be integers, at least one of them not 0. Prove that an integer e can be expressed as a…

A: Let a and b be intgers ateast one of them not 0. Our Aim is to prove that an integer e can be…

Q: Theorem 23. If u, v E Q then we can find integers a, b, d with d > 0 such that u = a and v = d'…

A: By Division Algorithm we know that,               Let a, b be two integers with b>0. Then there…

Q: II. Prove the following using Direct Proof. . If x and y are perfect squares, then xy is also a…

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Q: (2 3) Let A = Verify the Cayley-Hamilton theorem. Use the theorem to express 1 2 A² and A3 as linear…

A: The given matrix is A=2312. Cayley-Hamilton theorem: Cayley-Hamilton theorem states that, "Every…

Q: Let 6. 0 9 11 -2 1 3 4 8 A = 6 9 0 0 0 What is the rank of AT A?

A: Given matrix is A=650911-2001348000604000090000000

Q: 2.4.3. Let p(r1, T2) = 16, T1 = 1, 2, 3, 4, and r2 joint pmf of X1 and X2. Show that X1 and X2 are…

A:

Q: Let x, y, z, w ∈ Z and n be a positive integer. Suppose that x ≡n y and z ≡n w. Show that x + z ≡n y…

A: Suppose that x ≡n y and z ≡n w then, (i) we are assumed that  x-y=k1 n  and  z-w=k2 n  for some…

Q: Let n > 3 where x, y , z € R", |x – y| = d > 0, and r is some positive real number. Prove that if 2r…

A: It is given that x and y are d distance apart and r>d/2 where x, y, z belongs to Rn So locus z…

Q: 2. Using known theorems prove that P(ANB) > P(A) +P(B) - 1

A:

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: Prove that Σ = 2" . a. x=0 b. = 0. x=0

A:

Q: Prove that there are infinitely many solutions in positive integers x, y, and z to the equation x2 +…

A: We already know, (x,y,z) = (3,4,5) is a solution of the equation, x² + y² = z² i.e. 3²+4² = 5² Now,…

Q: Are the polynomials x? + 1,x? – 1, 2x – 1 linearly independent in P2?

A:

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: The set of polynomials {1, x² + 3x“, 2x² – 7x4} is a linearly independent set in P4.

A: NOTE: Due to time limit answer of 1 is given here, for other please ask in a different question.…

Q: Prove that if A ∈ Mn×n(F) has two identical columns, then det(A) = 0.

A: The determinant is the scalar value computed by expanding a row or a column.

Q: 2. Determine all k e R for which A is diagonalizable. [1 k 1 (a) A [1 0 k (b) A= |0 1 0 0 0 k

A: Solve the following

Q: 1. Let Z defined by the product of two numbers taken separately from two boxes containing 1, 2,3 and…

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer one question at a…

Q: Prove that P(A) - P(3)| 4 P(AAB). where A is the symmetric difference.

A: The following identity can be proved after breaking the LHS and RHS into mutually exclusive events,

Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: 5 Suppose Te L(V), m is a positive integer, and v eV is such that T-v0 but Tv=0. Prove that v, Tv,…

A: According to the given information, Suppose that:

Q: Prove that there do not exist a, b ∈ Z satisfying 54a − 39b = 2.

A: Solution:

Q: 4. The field of quotients of (Z2, +2)-2) is (a) Q2 (b) Z2 (c) Z4 (d) Zg

A: Field of quotient

Q: . Which of the following is the counterexample of: For all real numbers x, x≥1.  a. 12 b. 22 c. −12…

A:

Q: 2. Show that if x and y are perfect squares (that is, x = m? and y = n2, where m and n are…

A:

Q: There exists two odd integers whose average is odd. 1 1 1 There exists distinct positive integers a,…

A: (a) Consider two odd integers 2m+1, 2n+1 such that m+n is even. Now, Consider the average of the…

Q: A. Let R be a relation on set A. A={x:xeJ and 0 SxS 12), and R={(a,b): a-b is multiple of 2)…

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Q: 9. W is the set of all vectors in R2 whose components are rational numbers.

A:

Q: 5. Let a, b, N E Z where N > 0. Prove that [ab] = [[a][b], where [r] denotes the remainder of r…

A:

Q: (iii) Prove that there is a constant C such that 0(2º) 1.

A:

Q: Let A = {4n^2 + 1 | n ∈ Z}. Show that the size of A is equal to the size of Z≥0. Z≥0 is the set of…

A: A set A is said to be countable  if there is a bijective map between A and set of natural numbers .…

Q: 5. Let h (z) = (z+1)(z+6)(z+10) z(z+2)(z+3) (z+3)(z–2)°, (z+2)(z+6) What is the residue of h (z)at z…

A: Given a function on complex plane as hz=z+1z+6z+10zz+2z+3+z+3z-22z+2z+6 Residue at z=-2 is given by…

Q: - For a positive integer n, le Pn(r) = (x + 1)(r+2) (x + n). Show that the set of polynomials Pn (x)…

A: Here, we want to show that the set {pn(x)x+1,pn(x)x+1,...pn(x)x+1} is linearly independent. A set…

Q: [a Prove that I= { b. : a,b,c,d are even integers} is an ideal of M2(Z). d_ 1.

A: 1) The main aim is to show that I=acbd|a,b,c,d are even integersis an ideal in M2Z. to show that the…

Q: Prove that ||A||1 = max1<j<n{E1 |aij|}. m

A: We will be using the definition of norm. And will verify that the formula given for ||A||11 is…

Q: ) Prove that Вiт.m) - I(z)(m) for m >0 22'T(m +) 2

A: We use the property of Beta function and relation between Beta and Gamma function to prove the…

Q: Carry out the indicated algebraic operation. (If not possible, enter IMPOSSIBLE in any cell of the…

A:

Q: 2 17 1 12 7 15 2] What is the coefficient of X in the characteristic polynomial of the matrix A =…

A: Given matrix 5211127152Form a new matrix by subtracting λ from the diagonal…

Q: (c) Show that 1 and –1 are the only units that have finite order in R*.

A: Units in R*

Q: 82 핑 3 - 21 Prove that + Cn? m? Jyn=0

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Q: 2.7.5(1) Let R, and R2 be ordered fields that satisfy the Least Upper Bound Property, and let p: R1…

A:

Q: 1) Let F := {B E B(R) | B = –B}.Show that B is a o-field.

A: This is a question from Measure Theory.

Q: 5. Suppose n is a positive integer. (a) (b) Show that for any 0₁, not linearly independent. Rn. Show…

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Q: Let a, b, c, d be real numbers where c and d are not both zero. Let X = {x E R | cx + d + 0}. The…

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Q: 6. Inductively prove that B(0) = 0 for all n. Conclude that B(2) is C° function. %3D a

A: We prove the above theorem by induction. We have to prove that Bn0=0 ,∀n...1 Take n=0 then, Bn0=B0=0…

Q: 5. (a) Show that x2 – 5 does not have a zero in Q[/2]. (b) Prove that Q[/2] is not isomorphic to…

A: Zeros of the polynomial x2 - 5 are only sqrt(5) and -sqrt(5) which doesn't belongs to Q [ sqrt(2) ]…