16. (Calculus required) Show that the following sets of functions are subspaces of F(-∞, ∞). (a) All continuous functions on (-∞, ∞). (b) All differentiable functions on (-∞, ∞). (c) All differentiable functions on (-∞, ∞) that satisfy f'+ 2f = 0.

Q: Let R be the field of real numbers. Then {(x,x,x): x ER}is a subspace of R (R) ?

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Q: 3. Decide if the followin Let V and W be vector subspace of V.

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Q: 1. Let V be the set of all functions that are continuous on [-1.1]. Let W be the set of all…

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Q: Suppose that you wish to find the projection of u = (3,0,6)" onto the subspace W of R spanned by (2,…

A: We will find out the required solution.

Q: Let L2[0, 1] be all bounded Riemann integrable functions [0, 1] → R with inner product (f, g) Split…

A: Consider the provided question, Given that, Hn,kx=2n,    k2n≤x<k+122n-2n,   k+122n≤x<k+12n0,…

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Q: 3, 5x 60(x 2 + x;), 3 b) Y = In 7.

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Q: Show that the function f defined by 1, (x, y) = (1, –1) | f(x, y) : x2 + Y (x, y) + (1, –1) x + y is…

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Q: Let F = <5x^2, 2(x+y)^2, 9(x+y+z)^2> Find divF (divergence)

A: First of all, consider the vector F and find F1, F2, and F3.

Q: (-4,4) 1 -5 1 -2 (-1,0) -3 -1 -2 2 3,5

A: Note that we have common data only for (-4.4). For others, we don't have it. So please provide other…

Q: The flux of F out of a small sphere of radius 0.025 centered at (3, 3, 1), is 0.015. Estimate: (a)…

A: a)div F b)flux

Q: 6. Let C"(R) denote the sct of all real-valued functions defined on the real line that have a…

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Q: 4. Prove that a function which is harmonic everywhere on a plane and is bounded either above or…

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Q: 11) Is the set W = {f(x) E P(F): f(x) = 0 or f(x)has a degree n} a subspace of P(F) if n > 1?…

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Q: Show that f(x) = 1 x is not uniformly continuous on (0,1) but it is uniformly continuous on [1,2]

A: The function is continuous at x=a if, lim(x→a+)= lim(x→a-)=f(a)

Q: 6. Prove that an open interval (a, b) considered as a subspace of the real line is homeomorphic to…

A: An open interval a,b considered as a subspace of the real line is homeomorphic to the real line.

Q: 2. Let T, = |-5,2,,5 be a partition on [-5, 5] and 57 x > 0 Let f:[-5,5] –→ R be defind by f(x) = {3…

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Q: 9 Prove that if m(x) is differentiable on (-∞, 0) and its derivative m'(x) is bounded then m is…

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Q: 6. Consider FV, where V consists of all real-valued functions. Let W C V, where W = {f € V | |f(x)|…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: 4. Let S be the subspace of P₂ defined by S = span{1-x, 1+x, 5, 1-x², 2x}. Find dim (S), and show…

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Q: 4. UMNIA If the function h: [a, b] → R is bounded, Riemann integrable and satisfies h(x) > 0 for all…

A: Here given h:a, b→ℝ is bounded and Riemann integrable and hx≥0 for all x∈a, b, Then we have to show…

Q: 2. Determine the points where f is continuous where x + Y if (x, y) # (0,0) f(x, y) = x2 + y2 if (x,…

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Q: Let W be the subspace of the space of all continuous real- valued functions spanned by {cos2 t,sin2…

A: let U= cos2t,sin2t,cos2t and W be the subspace of all continuous real-valued functions spanned by…

Q: be the set Ol Che Imter that set of all differentiable real-valued functions f such that f' (1) =…

A: to Show W is a subspace we need to show zero vector is in W, W is closed under vector addition and…

Q: Determine whether the subset of C(-0, 0) is a subspace of C(-0, 0) with the standar The set of all…

A: we have to check for subspace

Q: Let X1, , X, be iid with population pdf given by h ( which integrates to 1 over the real line. Here,…

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Q: b) Is W = {f(x):f is a polynomial with degree <3,f (0) 0} a subspace of P: all 3 ? Explain your…

A: To see if the given subset is a subspace of the vector space of all polynomials of degree less than…

Q: (b) Determine whether A (2, 9) + (0, 0) (r,y) = (0,0) f(r, y) = %3D continuous or not.

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Q: Let C2(-∞,∞)={f(x) in C(-∞,∞)|f''(x) exists for all x} be the set of differential functions. Show…

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Q: Let S be the subspace of C[a, b] spanned by ex, xex, and x2ex. Let D be the differentiation operator…

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Q: Prove Fermat’s Theorem for the case in which f has a localminimum at c .

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Q: Fundamental Theorem of Calculus 2. Use the definition to show that f(x) = x² is Darboux Integrable…

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Q: Show that [0, 1] and (0, 1] as subspaces of R with the usual topology are not homeomorphic.

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Q: (A) a continuous function f that is defined on the entire ry-plane and has no absolute extrema on…

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Q: only 7) Suppose that U = E F x, y E F, where F is either C, R,or Q. One can y show that U is a…

A: We need to check only two conditions

Q: Let V be the set of all functions that are continuous on [−1.1]. Let W be the set of all functions…

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Q: Determine if the subset of C(- , 0) is a subspace of C(- ∞, 0). The set of all negative functions:…

A: We need to determine whether the subset of C(-∞,∞) is subspace of C(-∞,∞) or not. Here the function…

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Q: 7. Let V = Ca, b] be the set of continuous functions over [a, b], where a, b. Let a <d<b and let D=…

A: Given Ca,b be the set of continuous functions on [a,b] and a&lt;d&lt;b. To show that D=f∈V/f'd…

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A: Given function is fx=x−1+4x−2 (1) We have to find the limits of fx at the boundaries of its domain…

Q: Show that [0,1] and (0,1] as subspaces of R with the usual topology are not homeomorphic.

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Q: A company has $300,000 to spend on two products. Suppose that the Utility derived by the company…

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Q: 4. Which of the following are subspaces of F(-x, x)? (a) All functions ƒ in F(-x, x) for which f(0)…

A: The explanation of which of the following are subspaces of F(- infinity, infinity) is given in step…

Q: 4. Let S be the subspace of P2 defined by S = span{1-x, 1+x, 5, 1-x², 2x}. Find dim(S), and show how…

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Q: If X = {a, b, c} and 1 = {X, 4, {a}}. Then the subspace topo log y on A = {b} is {Ø, {a, b}} {Ø,…

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Q: For each of the following parts, determine whether the statement is trueor false. Justify your claim…

A: a) First put W to be a finite dimensional subspace which is generated by y = t. So, y is a solution…

Q: Let V = R and W = {x|x = (x1, x2, 1)}, Is W a subspace of V?

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Q: 12- Let X=C[0,1] and let W=[fo,f1,...] such that f;(x) = x', then we have W is A-an infinite…

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Q: The set of all continuous real-valued functions defined on a closed interval [a, b] in TR is denoted…

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Q: 3.) Determine whether the set of odd fuction: f (-x) = -f(x) which is a subset of C(-∞,0) is a…

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.