Find the kernel and range of each of the following linear operators on P^3 (the vector space of polynomials with degree less than 3): (a) L(p(x)) = xp′(x), where p′(x) is the derivative of p(x). Hint: Start with p(x) = ax2 + bx + c. (b) L(p(x)) = p(x) − p′(x).

Q: Compute the Fourier Transform of f(x) = e-ax-

A: Fourier transform of a function fx is defined as  Ffx=∫-∞∞fxeisx dx Gamma function =∫0∞e-x xn-1 dx

Q: The functions f₁(x) = x² and f(x)=x³ are orthogonal on the interval [-3,3] ? True False

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Q: Calculate the Fourier transform of f(x)= e if x>0 and f(x)30 otherwise

A: To find - Calculate the Fourier transform of fx = e-axif x > 00otherwise

Q: Find a linear differential operator that annihilates the given function 1+6x-4x3

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Q: Let f and g be functions defined by, f(x) = Vx+ 2 and g(x) = Vx2 + 1. Find the functions gof, , f +…

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Q: Show that the transformation T defined by T[x1, X2) = (2x, - 3x.2, Xq + 4, 5x,) is not linear. %3D

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Q: 3) a) Determine the linear Lagrange interpolation polynomial that passes through the points (-1, 2)…

A: Since you have posted multiple questions, we will provide the solution only to the first question as…

Q: Suppose f(x, y, z) = xyz + 2xy + 2² · Compute the following values -(2,8₁16) - (-2,-8, 16) (1,-1,0)…

A: We have to find functional value.

Q: 7) f(x)= xe-x 0 if -1<x<0 otherwise

A: Formula for the Fourier Transform of fx is  Fw=∫-∞∞fxeiwxdx  We are given fx=xe-xif…

Q: a) f(x) = {(3, –4), (2, 1), (–9,–3)} |

A: We have to find domain and range

Q: Verify the functions e3*, e-3* are linear dependent or Linear independent.

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Q: Let T: R? » P2(R) and U : R' → M2x2 (R) be lincar transformations. A student claims U must be…

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Q: Suppose T: R¹→P3 is a linear transformation whose action on a basis for R4 is as follows: -3 0 ਏਕ ਦੇ…

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Q: Determine whether the function defined by = 2x1y₁ - x1y2 - x2y1 + 2x2y2 is an inner product on R².

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Q: F: M2x2 → R, given by F(A) = det A; G: M2x2 → M2x2: given by G(A) = A'.

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Q: Determine the main base function as one of f(x) = x, f(x) =x², f(x) = Vx , or f(x) = -. 1 а) h(x) —…

A: A)h(x)=x-7The function h(x) is obtained byvertical transformations of 7 units downof f(x)=xThe base…

Q: Find the derivative of (Hайдите производную) (Баллов: 2) f (x) = x(-24)

A: Given,          fx=x-2x

Q: Prove that L(x) = (2x1 – x2, X2 + x3)" is a linear transformation.

A: To prove that L(x) = L( x1 , x2 , x3 ) = ( 2x1 - x2 , x2 + x3 ) is a linear transformation.

Q: Rewrite each pair of functions. (g) and g(t), as one composite function, f(g(e)). (If not possible,…

A: To find f(g(t))

Q: Use part one of the fundamental theorem of calculus to y = 3x + 7 t 1 + 1³ dt

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Q: Suppose that T is a linear transformation, with T(u,) = T(u2) =:: Find T(2u, - 3u,).

A: From the definition of linear transformation we can solve this question.

Q: Determine the domain of the following functions: За — 6 a) a(x) b) b(x) = /5 – x c) c(x) = 3x4 –…

A: We can find the domain of the given functions as below.