Assume that f:R² → R² is a continuous bijection such that the image of each line is a line. Show that f is affine.

Q: Let A = the set of straight lines in the xy-plane. Let a L b mean “line a is parallel to line 6”.…

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Q: Show that the into R² defined by T₁ = ( 9₁ ₂1₂) — 172,9₁) and T2: (F₁, F₂) → (Y¶₁₂ Y {₂) : are…

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Q: Let →a=a→= and →b=b→=. Find kk so that →aa→ and →bb→ will be orthogonal (form a 90 degree angle).…

A: We Know that, Two vectors u and v are said to be orthogonal vectors if and only if u . v=0 Here, a =…

Q: Let A be the set of all lines in the plane. Consider the relation a ~b if and only if line a is…

A: The reflexive relation compares the value of the object to itself. The reflexive relation for…

Q: ✓1.22 Represent each linear map with respect to each pair of bases. (a) d/dx: Pn → Pn with respect…

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Q: Show without using the Heine-Borel theorem that the line R is not a compact set in R ^ 2

A: The proof is done using the definition of compact set. A set S is said to be compact if and only if…

Q: What is the image of a horizontal line y = k € R under the mapping W = 2²?

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Q: 1 -2 4 17. Let a 4 a2 -3 and b : For what 7 value(s) of h is b in the plane spanned by aj and a2?

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Q: Find the the general solution of the differential equation (D² + 25)y= csc x sinæ + sin 5x cos 3x…

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Q: Let R denote the rectangle with corners (−1, −2), (4, −2), (4,1), and (-1,1), and let P denote the…

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Q: _et a₁ = -1 a₂= -7 -7 3 and b = 3 -15 h For what value(s) of his b in the plane spanned by a, and a

A: If b is in the plane spanned by a1 and a2 then b can be written as linear combination of  a1 and a2…

Q: Define a bijective map g : x2/3--->R.

A: Sol

Q: Let L₂ be the line passing through the point P₁=(-3, -1, -2) with direction vector -[0, 3, 3], and…

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Q: Let C be the line segment connecting the points P(−2,1,−4) and Q(5,−4,3). Evaluate ∫C(2x+4z)dy…

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Q: Assume (r1, y1) ~ (x2, Y2) → f(x1,Y1) = f(x2, Y2). Prove that ~ an equivalence relation.

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Q: Let x = (13, −20, 15), v1 = (1, −2, 3), and v2 = (−1, 1, 1). Then the set {v1, v2} is orthogonal. It…

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Q: Let l1, l2, and lz be nonconcurrent lines. Prove that there exist three points, exactly one each…

A: Given l1, l2, l3, three non cocurrent lines. the following cases are possible:  Case1: All lines are…

Q: Determine the values of y such that D(9,3,-2), E(3,4,7) and F(21,y,-20) are collinear

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Q: Let v = (3, 0, 0) and w = (0, 1, −1). Determine u = v × w usingthe geometric properties of the cross…

A: Cross product: more formally called the vector product a way of multiplying two vectors and produces…

Q: Let a, 1 1 -7 -7 3 and b = 3 15 h For what value(s) of h is b in the plane spanned by a, and a₂?

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Q: Suggest a "vector proof" of the fact that, inR2, two lines with slopes m1and m2 are perpendicular if…

A: Given: ℝ2 To Prove that : m1 m2=-1

Q: Let V₁ = -4 1 | -12 B-B- V2 = 4 -3 11 -5 " and y = For what value of h is y in the plane spanned by…

A: and We have to find the value of .

Q: Let a be a real number such that a opposit to 0 . An equation of the plane passing through A…

A: Given data:- Plane passing through the points, Aa,-a,-aBa,0,2aC-a,-a,-5a

Q: Let F = Vf and f = 5x²y - 5z. Calculate F. dr for the path r₁ = (t, t,0), 0 ≤ t ≤ 1. (Give your…

A: We will use fundamental theorem of line integrals

Q: Write a pair of curves that are homotopic and show that they are

A: it is known that two curves are said to be homotopic if one is deformed into another continuously.…

Q: If a is costructible,then evaluate that x=x and y=x are constructible lines. |

A: Claim :  If a is costructible, then x=a and y=a are constructible lines

Q: A relation R on a set A is called irreflexive iff (Vx E A)((x, x) ¢ R). It is called asymmetric iff…

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Q: Define the relation A=((x,x+ 1):xER] Show that A, is a function from R to R.

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Q: Let P(-4, 1, -3) be a point that is not on the line L . Let the line L pass-through points Q and R…

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Q: Show that every point of R is zero distance from Q

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Q: (d) Passing through the point (0, 1, 2), perpendicular to the line L and intersects L. L(xo. v) =…

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Q: Let R denote the set of real numbers. Let f: Rx R-> Rx R be a bijective function defined by f(x, y)…

A: Introduction :

Q: Sketch the domain D in the (x, y)-plane satisfying ≤ y ≤ 2e- and e ≤ y ≤ 3e. In your sketch, label D…

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