Example: Verify that each of the following mappings define a real inner product. (a) (...): R¹ × R¹ → R is defined as (x, y) = x¹y, for all x, y ER".

Q: Determine whether T is a linear transformation. T: F --->R defined by               T(f) = f( c),…

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Q: 5. Determine whether the function f : R+ → R defined by the rule f (x) = positive real numbers x in…

A: A function is said to be bijective if the function is both one one and onto. The given function is…

Q: Construct bijective maps (a) from [0, 1) to (0, 2]. (b) from R − {3} to R − {1} (c) from Z to Z −…

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Q: Let A and B be topological spaces and endow A x B and B x A with respective product topologies. Show…

A: Homeomorphism: Let X and Y be topological spaces; let f:X→Y be a bijection. If both f and f-1:Y→X…

Q: Show that X is Hausdorff if and only if the diagonal △ = {(x, x)|x ∈ X} is closed in X × X.

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Q: Let R[x] be a linear space with all polynomials p(x) = ax²+bx+c (a, b, c E R) whose degree is no…

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Q: Let f : R → R be defined as f(x) = x3 + 8 (i) Prove that f is a bijection (one-to-one…

A: Consider the given function.

Q: If f(x) and g(x) are arbitrary polynomials of degree at most 2, then the mapping (f,g) = f(-3)g(-3)…

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Q: Q;:Is the following mapping =|A + B| an inner product on M2(R).

A: No it is not a inner product.

Q: Let T : M2×2 → M2×2 be a mapping defined by T (A) = AT − A for A ∈ M2×2. (a) Show that T is a linear…

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Q: Prove that there is an additive function T : R --> R that is not linear.

A: Let T: V→W  be the linear transformation and we it is linear,…

Q: 1) Let X be a complex vector space and f2: X R be a linear functional. Define f:X C by f(x) = f2(ix)…

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Q: Let a, b, and c be distinct real numbers. Show that…

A: Consider the equation: We have to check whether it is a zero polynomial, symmetric and bi-linear or…

Q: Prove that the function f: R –→ R defined by f (x) = 3x – 5 is bijective.

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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: Write a [0.1]X[0.1] mapping to the tire surface .give a example.

A: Given That : [0.1]X[0.1]  To find : write [0.1]X[0.1] mapping to the tire surface also give a…

Q: is bijection. Do the same thing with Z,.

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Q: Let f(x)= x3 + x2 + x +1  ∈ Z2[x]. Write f(x) as a product of irreduciblepolynomials over Z2.

A: Given polynomial is: f(x)=x3+x2+x+1 Since, f(1)=0 So, x-1 is a factor. Now, factorize the…

Q: Give an example of a function f : (X1, d1) → (X2, d2) that sends compact sets to compact sets, but…

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Q: (a) What is the image of [1, 5]? (b) What is the image of [-3, 2]?

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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: C: Prove that the linear map T(x1, x2, x3, x4) = (2x2, 3x3, 4x4, x1) is an isomorphism of R4 to…

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Q: 2. Prove that ƒ : (R × R, +) → (R, +) given by f(x, y) = x + y is a homomorphism. Is g : (R × R, +)…

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Q: Consider the function f : R³ → R defined as V(x1, x2, x3) E R³ f(x1, x2, X3) = X1X2+ X1X3 + X2X3 a.…

A: We are given the function f: ℝ3→ℝ defined as ∀x1, x2, x3∈ℝ3    fx1, x2, x3=x1x2+x1x3+x2x3. For (a),…

Q: Let P₂ (R) be the space of all polynomials of order 2 and f: P₂ (R) → P₂(R) be the linear map…

A: We are given that f is a linear map.         f(1) = 1 + x        f(x) = 3 - x2         f(x2) = 4 +…

Q: Let L: R2 R³ be a linear mapping such that Find L -19 ([¹2]) -13 absin (a) 8 dr f 8 L D-||| (D-|| =…

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Q: Let V be a finite-dimensional inner product space over F. (a) Prove that the trace of every positive…

A: Below given clear explanation

Q: Define g : (-1, 1) → R by g(x) 1- x2 " Show that |(-1, 1)| = |R| by showing that g is a bijection

A: Concept Used: For a function f:A→B If f is a bijection, then |A|=|B| Consider the function g:(-1,…

Q: Define T : R? → R by 7((x, y)) = x+y. Describe the kernel of and fibers of T geometrically.

A: The kernel is a line with slope -1 and passes through the origin.

Q: Show that the mapping f:R2 → R² defined by f(x1, x2) 5x2) is linear. = (2x1 + x2, -3x1 +

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Q: 1. (a) Show that the function f : [0, 1) → [0, ox) defined by f: xH 1- x is a bijection. (b) Find a…

A: Given, f:[0,1) →[0,∞) f(x) = x1-x

Q: Let V be the set of all pairs (x, y) of real numbers together with the following operations: (x₁,…

A: V be the set of all pairs (x,y) of real numbers  together with the following operations ;…

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: Consider the function g : ZxZ → Z × Z defined by g(x, y) = (4xy, 9x + 5y). (a) Is g injective? (b)…

A: (a) we see (5, 0) and (0, 9) are different elements in ZxZ.      But, g(5, 0) = (4.5.0, 9.5+5.0) =…

Q: Let M(R) denote the set of all mappings from R to R. For mappings f,g: R → R, define pointwise…

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Q: Prove or disprove whether the function f (r) = a +2 is a bijection from R R %3D

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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: Let M(R) denote the set of all mappings from R to R. For mappings f, g: R → R, define pointwise…

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Q: 2) Find the indicated norms of the following linear functionals defined on RS b) f(x1,X2, X3, X4,X5)…

A: This question is about application of functional analysis