(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) = (a+b√−5)(a − b√−5) = a² + 5b². Show that if x, y = R*, then N(xy) = N(x)N(y). (b) Show that u € R* is a unit if and only N(u) is a unit in Z. (c) Let U(R) be the set of units in R. Show that U(R) = {−1, +1}.

Q: Let X {1,2, 3} and Y {3,4}. Define f : X → Y as follows: = f(2) = 3 and f(3) = 4. Draw an arrow…

A: We will draw the arrow diagram for function mapping from set A to set B.

Q: The relation on Z × N defined by, (a, b) ~ (c, d) if ad bc.

A: The objective is to show that the relation a,b~c,d then ad=bc is an equivalence relation. To show…

Q: Let T: R2 - R3 be the linear map which satisfies and 3 Give the formula for T. x + 3y x + y

A: We have to find

Q: x, y, z)=x²√√x - y² - z²

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Q: C: 1. Is there a linear map from R2 to R³ such that T(1, 3) = (0, 0, 0) and T(-2,0) = (0, 0, 0) and…

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Q: Given a linear map L:R-R such that L(1,0,0)=(3,-1),L(0,1,0)=(2,1),L(0,0,1)=(3,0) , then find…

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Q: Consider the the linear maps ƒ : R² → R² and g : R² → R² given by f((x, y)) = (x − y, 2 + y) and…

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Q: The map f:R³ R³ defined by f(x.y.z) = (x+y+z.0.2) is a linear map. True False

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Q: Suppose that the translation T is described by (?,?)→(?+2,?−1)(x,y)→(x+2,y−1). What translation…

A: We have to find the translation described in x,y→x+2,y-1.

Q: Find a ≤ b such that

A: ∫abx2-9dxApply the Sum Rule:…

Q: f: R² → R2 with f(x, y) = (2x, x+y) Which statement is correct? f is NOT a linear map and it is not…

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Q: g is a coordinate system for a line L and a ∈ R, then the function f : L → R given by    f(P) =…

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Q: -2y 2x (x²+y²) ²³ (x²+y²)² Compute fF.dr, where F(x, y) = oriented counterclockwise. (Use symbolic…

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Q: Show that the function (, -): R² × R? → R defined by setting = 1122+ 2x12 + 2y172 + 8y142 = is an…

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Q: 4. Consider the the linear maps f: R² R² and g: R² R² given by f((x, y)) = (x-y,2+ y) and g((x, y))=…

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Q: Solve the initial value problem 16y" – 24y' + 9y = 0, y(0) = 3, y' (0) = 5.

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Q: Write a [0.1]X[0.1] mapping to the tire surface .give a example.

A: Given That : [0.1]X[0.1]  To find : write [0.1]X[0.1] mapping to the tire surface also give a…

Q: Let X be the set {1, 2, 3, 4, 5, 6}, Y the set {a, ß, 7, 6}, and ƒ the map f(1) = a f(2)= a f(3) = a…

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Q: Let X = {a, b, c, d,e} and let T = {X, Ø, {a}, {a,d}, {a,e}}, then * O (X,T) is not connected but it…

A: No information is given for topology on Y So i have solved only the middle question

Q: (a) What is the image of [1, 5]? (b) What is the image of [-3, 2]?

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Q: C: Prove that the linear map T(x1, x2, x3, x4) = (2x2, 3x3, 4x4, x1) is an isomorphism of R4 to…

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Q: 157 Find the linear mapping f(z)=9| e 4 z+8i applied to the circumference | z-i |-7. 2

A: We know that eiθ=cosθ+isinθ ∴ei5π4=cos5π4+isin5π4             =cosπ+π4+isinπ+π4             =-12-i12…

Q: Which of the following maps from R² to R³ are R-linear? 2 f(x, y) = (x. x - 2. x+y+z) f(x, y) = (x…

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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: 2) The contour map of f(x, y) is shown below. Use it to find fx (1,3) 10 10 8 8 6 =2 6 0 x=2 6 8 10

A: fx​(1,3)=1   Explanation:From the given contour map, we can see that, f(1,3)=4  Similarly we can…

Q: Let f(r, y) = |r|+ lyl- (a) Draw three r-cuts of f(r, y), three y-cuts of f(x,y), and one z-cut.…

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Q: Show that the map T: R³ R² given by T(X₁, X₂, X3) = (X₁ + X₂, X3) is an oper map.

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Q: please solve this problem

A: Consider the provided question,

Q: 8. Show that the mapping T: V,(R) → V½(R) defined = (3a, - 2az + az3, a1 – 3a, – 2az3) is a linear…

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Q: Graph the preimage and image of AABC with A(2,1), B(0,-1), and C(-3,2). Translate it by…

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Q: True or False: The linear map T:P 3(R)->R^4 defined by Tp= (p(1).p(2).p(3).p(4)) is invertible. O…

A: Given: T: P3R→R4 defined by  Tpx=p1, p2, p3, p4T1=1,1,1,1Tx=1,2,3,4Tx2=1, 4, 9, 16Tx3=1, 8, 27, 64

Q: Need only handwritten solution only (not typed one).

A: The given set .To find the image of the set  under the transformation .

Q: The function f is defined as follows: f: R² R² with f(x, y) = (2x, x+y) Which statement is correct?…

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Q: We consider a function f from R to R such that f(0)=5. Show that f cannot be a linear map. Hint:…

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