If R be the additive group of real numbers and R+ the multiplicative group of positive real numbers show that the following mapping are isomorphism_ (i) f: R→ R+ s. t. f(x) = ex, x € R. (ii) f: R+ → Rs. t. f(x) = log x, XER+.

Q: Let S : RR be defined by S(x1, x2, 3,...) = (T2, T3, T4,...) (a) Using the definition of linear…

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Q: A transformation T: R² → R² is defined by T () = + 12 The image of the curve f (x) =-- - 3 under the…

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Q: `

A: Certainly! Let's break down each part for a better understanding: 1. Maximum dimension for the image…

Q: of the following linear transformations are isomorphisms? -2x-6z T R R where T y y-z 2x+6z ab T2…

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Q: 2. Let G = (R*, .), the multiplicative group of nonzero real numbers. Let H = {x ER: x -1} be the…

A: 2. Let , the multiplicative group of non-zero real numbers.Let  be the group with operation .Prove…

Q: 15. Let G be a group and let g be in G. Define a function :G → G by ,(x) = gxg Show that :G→G is…

A: Let G be a group and let g be in G . Define a function ϕg : G→G by ϕgx=gxg-1. To prove: ϕg : G→G is…

Q: If U:M3x2(F)-->F6 is a linear transformation, then U is an isomorphism.

A: True.

Q: The Galois group of  x4 +x2 +1 is the same as that  of  x6 -1 and is of order 2.

A: Since the primitive nth root of unity is        e2πi/n=cos2πn+i sin2πn. Then primitive third root…

Q: Which of the following properties are true for the transformation T : R° → R² wit x – 3y + z — х +…

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Q: Let T : M2×2 → M2×2 be a mapping defined by T (A) = AT − A for A ∈ M2×2. (a) Show that T is a linear…

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Q: Chapter 2, exercise 5.4: Let ƒ : R+ → C+ be the map f(x) = e¹ª. Prove that f is a homomorphism, and…

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Q: Let T: P2 → P2 be defined by T(p(x)) = xp'() A. Describe the kernel of T. B. Describe the range of…

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Q: Let f: Z/4Z → Z/2Zx Z/2Z be defined by f([0])= ([0], [0]), ƒ([1]) = ([1],[1]), f([2]) = ([0], [1]),…

A: We see,         [2] = [1] + [1]    but, f([2]) = ([0], [1])          f([1]) + f([1]) = ([1], [1]) +…

Q: Let L be a linear Prove that dim[Range (L)] + dim[Kernel (L)] = dim[V] transformation from V W

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Q: In the frieze group F7, show that zxz = x-1.

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Q: JUNa : SA SB such that for o E SA f: A B be a 1-1 function mapping onto B. Define (0)= E SB such…

A: We are given that f : A-->B be 1-1 function mapping onto B. In other words, f is one-one and…

Q: (i.) Let F(x)= x². F is an automorphism on (R+,.) (j.) The set of non-zero real numbers with…

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Q: 7. Which of the following types of linear transformations are always diagonalisable? (a) The…

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Q: Let T: RR∞ be defined by T(x₁, x2, 3,...) = (0, x₁,x₂,...) (a) Using the definition of linear…

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Q: Consider the following functions. (c) f: GL₂(R) → RX, f + ((ad)) (($)) (d) f: GL₂(R) → R, ƒ (e) f:…

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Q: Prove that the mapping f : U(16) ---> U(16) given by f(x) = x^5 is an automorphism.

A: The given map is f:U16→U16 defined by fx=x5. We have to prove that f is an automorphism. We know…

Q: Show that the determinant gives a group homomorphism det: GL(n,R)->R*, where R*=R-{0} under…

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