(2) If a, b = Z, we set N(a+b√-5) = (a + b√−5)(a − b√-5) = a² + 5b². Let R = Z[√5] and R* = R-0. Consider the nomm map N : R* → Z*. (d) Show that 2,3,1+√-5, 1-√-5 are irreducible elemetns in R. (2) Observe 6 = 2.3 = (1+√√-5). (1-√-5). Show R is not a unique factorization domain.

Q: Construct bijective maps (a) from [0, 1) to (0, 2]. (b) from R − {3} to R − {1} (c) from Z to Z −…

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Q: ) The linear transformation T(x1, x2, x3, x4) = (x₁ - x₂ + x3 T is given by the following formula…

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Q: For 3 and 4given the translation rule: T(x,y) (x+2,y-3)

A: Find the image of (5,-1) and preimage of (-6,7)

Q: Consider the following linear map T: P3 → P3 defined as follows: T(ao + a1x+ azx² + a3x³) = ao +…

A: For T to be an isomorphism we must check that it is one one and onto both. And we can check it by…

Q: Use the function to find the image of v and the preimage of w. T(V₁, V₂) =(√² V₁. √2, - V2, V1 + V₂,…

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Q: (2) Let R = Z[√5] and R* = R−0. Consider the nomm map N : R* → Z*. If a, b = Z, we set N(a+b√-5) =…

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Q: 5. Let T: P₂ (R) → P₂ (R) be defined as P(ax² +bx+c)=((2a − b)x² + (3a + 3c)x+ (a − 2b - 3c)). Find…

A: 5. Given that T:P2ℝ→P2ℝ be defined as Tax2+bx+c=2a-bx2+3a+3cx+a-2b-3c. We have to find the KerT and…

Q: Suppose a transformation has a mapping as follows:(x,y)→(1/2 x + 3, -2y + 4). If this mapping were…

A: Given: A function y=x and a transformation mapping x, y→12x+3, -2y+4.

Q: Let R(s, t) = G(u(s, t), v(s, t)), where G, u, and v are differentiable, and the following applies.…

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Q: A: pp. A: px p. . A: pp(3)x3 + p(1)x.

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: 2. Is there a linear map from R³ to R³ such that T(1,3,0) (0,0,0) and T(-2,0,0) = (0,0,0) and T(3,…

A: We  have  to  find a  linear  transformation T: ℝ3 →ℝ3   for  which…

Q: Example 9. Classify each relation as constant, linear, quadratic, or neither. a) y = x² + 5x + 6 b)…

A: The solution is given below:

Q: Supposer is an invertible function with domain [1, 5] and range [2,7]. Then the domain of r−¹(x − 1)…

A: Given that r is an invertible function with domain [1,5] and range [2,7].It means that for every…

Q: 2. Let : RR and y: R+ → R be mappings with 6(x) -2e-x. (a) Find the range of y. = 1. Use your answer…

A: Here's a summary of the answers:  Solutions:(a) Find the range of γ (b) Use the definition to show…

Q: O (X, y) → (x – 6, y + 1) O x, y)(x-6, y- 1) O X, y)(x+2, y+ 1) O (X, y)(x+6, y- 1)

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Q: |-2+1 () 7:2 だ,て メ-ドコ X, +1メ」 メレ-3x, transformation is gi ven t of trensformation; 4) Show that it…

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Q: For 16 - 19 consider the translation defined by the coordinate notation T(x,y) (x+4,y-6)

A: Explanation of the answer is as follows

Q: 7. Compute gfg, the conjugate pair of f by g for f= (2, 3, 5, 4) and g = (1, 3, 2)(4, 5).

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Q: 5. Given f(x) = -2/4x – 8 +1 a. Determine the parent function b. Determine the mapping rule c. Map…

A: Given function

Q: 2. Let us define a. Verify that f(x, y) = af ду xy(x² - y²) x² + y² af y(x² - y² + 4x²y²) əx (x² +…

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Q: Let A B and g: B-C be maps. Prove that if g of is one-to-one, then / is one-to-one.

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Q: 2. Let us define a. Verify that f(x, y) = c. Now consider the function git Similarly, consider the…

A: Given Data: Let us consider the given function is fx,y=xyx2-y2x2+y2

Q: II. Let V =span- and define the transformations a b T [a² – b? c? – d²] ; S(A) = adj(A) P Determine…

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Q: 4) Let ƒz = dƒ/əz and ƒz = 0ƒ/əz. Recall ƒ = u + iv, the notation f is used for the complex…

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