Show that the following functions define an inner product on (R)2 where U= (x,y) and V= (x2,y2) a) = 3x1x2 + y1y2 b) = (x1x2)/2 + (y1y2)/4

Q: Determine if the function defines an inner product on R², where u = (u, v) = ₁V₁ satisfies (u, v) =…

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Q: Find (u, v), ||u||, ||v||, and d(u, v) for the given inner product defined on Rn. U= (0, -3), v =…

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Q: Show that D^2 = {(x, y) ∈ E^2: x^2+y^2 ≤ 1} ⊂ E^2 and the space containing a single point are…

A: In algebraic topology, two topological spaces are said to be homotopy equivalent if there exists a…

Q: 9. Show that the function (-, -) : IR² × R² → IR defined by setting = #1#2 + 2x12 + 2y1*2 + 8y1Y2 /2…

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Q: Find (u, v), ||u||, ||v||, and d(u, v) for the given inner product defined on R". u = (5, 0), v =…

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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: \(d\) is a metric on \(X\). Show that \(\rho : X^2 \to [0,\infty)\) defined by \[ \rho(x,y) =…

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Q: Use the functions f and g in C[-1, 1] to find (f, g), ||F||, ||9||, and d(f, g) for the inner…

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Q: Let S = Þ (D), where D = {(u, v) : u² + v² ≤ 1, u ≥ 0, v ≥ 0} and Þ (u, v) = (2u+1, uv, 3u + v). (a)…

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Q: Let T: R3 -> R2 be defined by T(a, b, c) = (a + b, 2a —- c). DescribeT-1(1, 11).

A: T: R3 -> R2 Hence, T-1 : R2 ----> R3 So, necessarily we have to find a, b & c such that…

Q: True or False: (p, q) = p(1)q(1)+p(2)q(2) +p(3)q(3) defines an inner product on P2(R).

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Q: Justify that (•, ·) defines an inner product on R?. the function (', ·) defined on R? as follows: U1…

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Q: 6. Show that each of the following are not inner products on R³, (a) (u, v) = U₁V₁ + U2V2; (b) (u,…

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Q: Explain why is not an inner product for u =(u1, u2) and v =(v1, v2) in R2 when = u1u2 + v1v2 using…

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Q: 2. A robot performs the following operations on a set of points (x,y) in the 2D plane to check the…

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Q: Let f be a function that takes elements of R?rR? into R' as follows: f(u, v) = |u1v1|+ |u2v2|, where…

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Q: f: X → Y and g: Y→ V be bijective functions. Show the following: ) gof: X → V is injective. gof: X→V…

A: It is given that f:X→Y and g:Y→V are bijective functions.