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: 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: (b) Let R = {(x, y) E R × R : 1 < ¤ < 4 or x = 6}, and S = {(x, y) E R × R : -1 < y < 1}. i. Find…

A: Solution: The given relations are R=x,y∈ℝ×ℝ : 1≤x≤4 or x=6 and S=x,y∈ℝ×ℝ : -1&lt;y&lt;1

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: Let r¡(t) = (-4,0, –5) + t(0, 1, –3) and r2(s) = (2, –2, 10) + s(2,0, 3). Find the point of…

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

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Q: Find the closure of a straight line in R x R_.

A: Given: Straight line in ℝ×ℝ. To find: Closure of straight line in ℝ×ℝ.

Q: Show that the line [x, y, z]= [5,-1,3]+[4,-5,2] lies on the planes 3x+2y-z = 10 and 7x+4y-4z=19.

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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: Consider the points P = (1,2,0), Q = (0,1,2), and R=(2,0,1). Let A be an arbitrary point in…

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Q: Find the plane through P = (4, −1, 9) containing the line r(t) =〈1, 4, −3〉+ t〈2, 1, 1〉.

A: The plane contains the line r(t)=1, 4, -3+ t2, 1, 1, so a vector in the plane is 2, 1, 1. The line…

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

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Q: Let l1 : (x, y, z) = (131, 89, −83) + r (42, 28, −28) where r ∈ R and l2 : (x, y, z) = (29, 21,…

A: LetandShow that the lines  and  are identical.

Q: If ů, = |8|,02 = 5,03 show that {v1, v2, v3} is linearly dependent in R', and determine the linear…

A: Letv1=2i^+8j^+3k^v2=i^+5j^+2k^v3=i^+j^Let…

Q: 1. L: V3 V4 defined by L(a, b, c) = (a - b,b+c-d, a+b-2d, b-c-d) 2. L: V4 → V3 defined by L(a, b, c,…

A: According to Bartleby guidelines we can only solve 1 question or three subparts. So put other one…

Q: Evaluate v (x? v + x y z²) · dr where C is the line segment from (1, – 1,0) to (2,1,3) | Select one:…

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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: 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: 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 l,, l2, and l3 are nonconcurrent lines, prove that there exist three points. Exactly one each…

A: Three or more points are said Lto be collinear if they all lie on the same straight line If l1 , l2…

Q: Let C is the line segment from (0, 0) to (2, 1). Then ydx + xdy is 2 01 0 None of the above.

A: Step 1:Step 2:Step 3:Step 4:

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: 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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