Let X =  R x R  x R = R3 and define the metric d: R 3 x R 3 to R  by d((x1, y1, z1), (x2, y2, z2)) = |x1 - x2| + |y1 - y2| + |z1 - z2| Describe the neighborhood N(0, 1) where 0 is the origin in R3.

Q: (a) Prove that f is a one-to-one correspondence. (b) Prove that ray AF is between rays AB and AE if…

A: (a) To prove that f is a one-to-one correspondence, we need to show that f is both injective and…

Q: b) Is there a linear map T: V → W, where dim V = 3, dim W = 4, rank T = 2, and dim ker T = 1?

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Q: Determine the intersection if any between the line L₁: [x, y, z] = [3, 2, − 5] + r[0, 1, 4] and the…

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Q: The surfaces intersect in a space curve C. Determine the projection of C onto the xy-plane.…

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Q: Consider the following mappings on RXR: * d1(1, y) = e²=y, d2 = |2r - 2y). and d3 = V12r – yl. %3D…

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Q: Let X=R2 and let d be the Euclidean metric.  Define d3:  R2 x R2 to R by d3(x, y) = min{1, d(x, y)}…

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Q: Let (X, d) be a metric space. Define f: Xx X → R by f(x, y) = = that f is a metric on X. d(x, y) 1+…

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Q: Let (t) = (t, t, tº, tª) for all t € R. Define C = Þ(R). Let p = (1, 1, 1, 1). Derive the tangent…

A: Let  for all Let the point be We need to calculate the tangent and normal spaces to C at p.

Q: 1.T.2 Let V be the set of all twice-differentiable functions f : R → R satisfying f"(x) = f(x) – 5.…

A: Let, f and g be two functions in V.  Then,        f"(x) = f(x) - 5       g"(x) = g(x) - 5.

Q: 7. Consider the ordered bases S and T of R: ()} S = {81, 82, $3} and T = {t1, t2, t3} = We define a…

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Q: Let f function from (R,d) to (R,d1) defined as d(x) = x, where d is the usual metric and .di is the…

A: Metric space function