(a) Suppose y is the set of all even functions in C[-1, 1]. Find the orthogonal complement y¹ of y in C[-1, 1] under the inner product on C[-1.1] given by (x,y)= £₁x(1) y(1) d (C[-1. 1] - the space of all continuous real valued functions on [-1, 1]).

Q: Suppose f and g are non-constant, differentiable, realvalued functions defined on (−∞, ∞).…

A: We plug x=0 and y=0

Q: Let fi =1+2r– 2r², f2 = 2+3r – 4r², f3 = 1+4x – r² and set B = (f1, f2, f3). T:R2[r] → M2×2(R) is a…

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Q: 6: F7 → F2 where (x1, x2,...,x7) = (X1 + X2, X3 + x7). Prove is not a linear functional on F7

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Q: (a) Prove the set (A(*)./a(*),fa(a)) is linearly independent. (b) Use Gram-Schmit process to find an…

A: Given,     f1(x)=3,  f2(x)=1-x, and f3(x)=x2+2x-1

Q: 1. If f is the map that sends each complex number z = x + yi → y x Show that f(z1z2) = f(z1)f(z2),…

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Q: Let C = {0, 1, 2, 3} and f: C → C, g: C → C with ƒ = {(0, 2), (1, 2), (2, 3), (3, 0)}, {(0, 3), (1,…

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Q: The function f(z) is complex valued in the complex plane. f(z) is bounded and entire. Prove that…

A: As per the Louisville's theorem, every entire function which is bounded must be constant. Use the…

Q: Let f be the function from {a, b, c} to {1, 2, 3} such that f (a) = 2, f (b) = 3, and f (c) = 1. Is…

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Q: Let G : R? → R³ and F : R³ → Rª be defined by G( (x, y)) F( (a,b, c) ) (x +y, x – y, 2y) (a + b+ c,…

A: Given G:R2→R3 is defined as Gx,y=x+y,x-y,2y Also,  F:R3→R4 defined by Fa,b,c=a+b+c,2a+c,b+2c,a+b.

Q: Let P3(R) be the space of polynomials in r with real coefficients, having degree at most 3. Let T :…

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Q: Let f: R² → R be defined by f((x, y)) = 8x +9y. Is f a linear transformation? a. f((x1, y₁) + (x2,…

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Q: 1. Define f : (- o, 2) → (- 0, 1) by f(x) = . Show that f is bijective. x+1 *-2

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Q: Prove that the following functions are bijections: (a)  f: R → R given by f(x)=ax+b, where a ≠ 0.…

A: Let f:A->B be a function. Injective function: f is said to be injective if different elements…

Q: af Ꭷf f(x, y) — f(y, x) = (x − y) — (x, y) + (y − x) — (x², y") ду or some point (x*, y*) € R² b)…

A: aSince given partial differential equation is               fx=fy⟹      fx-fy=0Auxillary equation…

Q: 11) Let f: IR4 → IR³ be defined by : IV f(x₁y₁z₁t) = (2x+y-t, x+8y-92- 2t, x-2y + 3z) (a) show that…

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Q: 2 Let f : R ² → R be defined by f((x, y)) = 6y – 9x. Is f a linear transformation? a. f ((x₁, y₁ ) +…

A: The objective of the question is to determine whether the function f, defined by f((x, y)) = 6y –…

Q: Prove that the function : R-{1} →R- {2} defined by 2a f(x)= s bijective. 2 1

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

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Q: Q2: Let T₁, T₂, be the following bounded linear operators ¹ → T₁(x₁, x₂. x3,) = (x₁, X₁, X₁, X₁,…

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Q: (b). Let Y = {a, e, o, i, u}, take two relations R and S on Y and Prove that (SoR)-1 : R-1os-1.

A: composition of relations

Q: Let T be the linear functional defined on C [a b] (with sup norm) by T (x) = ex (t) dt for x eC[a,…

A: Since T is a linear functional defined on        C [a,b]

Q: a) Let f: R - [1, co), where f (x) = x2 + 1. Determine whether the function maps onto the given…

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Q: Show how a real

A: To Prove that how a real linear functional u on a complex linear normed space gives rise to a…

Q: f: R →R by f(x)= x^2 and V is a subset of the codomain. Find a set V such that V has only one…

A: We are given that f : R → R by f(x) = x2 and V is a subset of the codomain. We need to find the set…

Q: Prove that the function f: R W {2} R {5} defined by - 5x + 1 f(x) = #-2 is bijective.

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Q: Prove where the following complex function is analytic

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Q: 4) Let ƒz = dƒ/əz and ƒz = 0ƒ/əz. Recall ƒ = u + iv, the notation f is used for the complex…

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Q: X2 - X1' Let L:R3 → R³ be a linear operator such that L ×2 ) =| X3 - X2 , then Ker(L) spanned by: X3…

A: we know that Ker(L) is solution of the system L(X)=0  Reveal sold the system and find the ker(L)…

Q: Example: Show that the function Az) = z² is analytic. z = x+iy S(2) = z

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Q: Let f, g and h be R- R functions, f(x) = 2x - 1, g(x) = 3x+1, and h(x) = x +1. Compute the following…

A: I will solve all three subparts for you and provide detailed answers.