Prove that if gcd (a,b)=1>then gcd (ab,ab)=l

Q: Are x, sin(x) and cox(x) linearly independent

A: True x, sin(x), cos(x) are linearly independent

Q: Exercise 2. Show that if B + Ø then dom(A × B) = A. %3D

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Q: Show that [x, p;] = iħ.

A: note :  As per our company guidelines we are supposed to answer ?️only one question. Kindly repost…

Q: Show that MnXn and Rn^2 are isomorphic.

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Q: Prove that det(AB) = det(A) · det(B) for any A, Bϵ M2x2(F).

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Q: Show that if gcd (a, y) = 1 and a \ xy Then a\ x

A: Definition : Greatest common divisor ( gcd): Let  a and b be are two  integers, not both 0.  A…

Q: If B is invertible, prove that det(B-'AB) = det(A).

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Q: Prove that fVf · dr = 0.

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Q: If A²=0, prove that A cannot be invertible.

A: Given,If A²=0, prove that A cannot be invertible.

Q: Let A = and v= Verify A(u + v) = Au +Av. 2.

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Q: Prove that S4 is not isomorphic to D12.

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Q: If xly then gcd(x,y) = y. إختر واحدا:

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Q: Let f(T) and g(T) be polynomials in Q[T] whose gcd in Q[T] is 1. Show that 1 is also their gcd in…

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Q: Show that there exist m and k such that ma + kn = 1if gcd(a, n) = 1

A: Let gcda,n=1.

Q: Prove directly. a) [(Vx (p(x) → q(x))) ^ (3x p(x))] =3 x q(x)

A: When we prove logical equivalence, we use the truth table. But sometimes, the truth table becomes…

Q: Prove Theorem| 2.3: (i) (A+ B)" = A" + B", (i) (A")" = A, (iii) (kA)" = kA".

A: To prove the given transformations

Q: Exercise 3. Show that f : Z → N so that f (z) = z² + 1 for all z E Z is neither one-to-one %3D nor…

A: One-One Function: A function is said to be One-One if for every element in the domain there exists…

Q: Prove or disprove that 0 ≤ x2 - xy + y2 for all x and y in Z.

A: Consider the given inequality 0≤x2-xy+y2  for all x and y in  ℤ Now,…

Q: 2. Prove that if 5BA+ ACA – I = A, then A must be invertible.

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Q: Show that if gcd (a, y) = 1 and a \ xy Then a \ x

A: if 1. gcd(a,b)=1then there exist some integers m and n such that am+bn=12. suppose a and b are…

Q: Let E Os 0s n, 0s zs 1, 8Vzs rS 8. Compute

A:

Q: Show that for all z1, Z2 E C, Then we have: |Iz1| – |z2|| < |z1 +z2l

A: 2) The inequality can be proved as follows:

Q: Prove that ||A||1 = max1<j<n{£", |aij|}. %3D

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Q: b) Prove that if x <y and z<w, then x+z<y+w.

A: Given: Suppose x&lt;y and z≤w. To prove: x+z&lt;y+w.

Q: Exercise (7): Find f, zdxdydz on [0,1] x [0,2] x [0,3]

A: Find: ∫∫∫Vzdxdydz on 0,1×0,2×0,3

Q: Exercise 2. Prove that if A = B and C z D, then A × C = B x D.

A: Given data: A≈B......(1)C≈D......(2) The objective is to prove A×C=B×D

Q: Theorem 2. Let a, b, c E Z such that gcd(a, c) = gcd(b, c) = 1. Then gcd(ab, c) = 1. %3| Proof.…

A: the gcd of two integers is the largest positive integer that divides each of the integers.

Q: Let f(x, y, z) = x³ – yz?. Find af af of and ax' dy az

A: Given : fx,y,z = x3-yz2

Q: = Prove that E = ehd.

A: We know that Ef(x)=f(x+h)

Q: Prove that if A ϵ Mnxn(F) has two identical columns, then det(A) = 0.

A: Given: A∈Mn×nF i.e. A is a matrix of order n×n.   Consider a matrix with first two column as…

Q: 6. Prove that if gcd(a, b) 1, then gcd(a +b, ab) = 1. %3D %3D

A: Given that gcd(a,b)=1 If possible , let us assume that  gcd(a+b,ab)=d,d≠1 Then the number d must…

Q: In Let x, y ER" be given. (a) If ||x|| = ||y|| = 1 and x - y = 1, show that x = y. (b) If ||x|| ≤ 1…

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Q: Prove that if gcd(a, b) = 1 and cla, then gcd(b, c) = 1.

A: We need to prove If gcd(a,b)=1, then gcd(b,c)=1. The proof is given below.

Q: Show that if gcd (a, y) = 1 and a \xy Then a\ x

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Q: Prove that if gcd(a, b) = 1 and gcd(a, c) = 1, then gcd(a, bc) = 1. %3D

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Q: If x]y then gcd (x, y) = y. إختر واحدا: Las O

A: Given x|y hence y=ax It implies that y is a multiple of x

Q: Prove Theorem  A + (B + C ) = (A + B ) + C .

A:

Q: 4. Assuming that gcd(a, b) = 1, prove the following: (a) gcd(a +b, a – b) = 1 or 2. %3D %3D

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: Let X, Y ~ U[a, b] be independent. Define Z = max{X, Y}. Find the cdf of Z.

A: Let X, Y ~ U[a, b] be independent.

Q: Consider a LU factorization of 3 A = 6 as A=LU. Then L2,1 and U1,1 are respectively equal to

A: solution:-

Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

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Q: Prove that gcd(a + b, a - b) = gcd(2a, a – b) = gcd(a + b, 2b). Show your work! Justify each step.

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Q: Let a, b ∈ Z, not both zero. Prove gcd(−a, b) = gcd(a,−b).

A: We have gcd(-a , b)=gcd(a,-b) let gcd(-a , b)=d Where -a=de  , b=df  ⇒a=d(-e) , -b=d(-f)⇒d is…

Q: Let a = (– 1, 5, 0) and b = (3,0, – 1). Find the component of b onto a.

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Q: Prove that (aA + bB)t = aAt + bBt for any A , B E Mmxn(F) and any a,bϵ F.

A: Consider, aA+bBt=aAt+bBt    ∵A+Bt=At+Bt=aAt+bBt     ∵kAt=kAt for any scalar k.=aAt+bBt

Q: (a) Verify the Cayley-Hamilton theorem. (b)  Use the theorem to express A2 and A3 as linear…

A: Part (a), Let, A=4-112 Now, the characteristic equation of A,…

Q: z, Z2 Prove

A: If z =x+ iy then  z = x- iy. i2 =-1.

Q: Prove that if A C R and t > 0, then |A| = |AN(-t,t)|+|AN (R\(-t,t))|-

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