p(1) 3. Let T : P3 → R³ be defined by T (p(x)) = p(2) p(3) Show that T is a linear transfor- mation, and find ker (T). Recall that P3 is the space of all polynomials of degree at most 3, P3 := {p(x) = ao + a1x + a2x² + a3x*|a1, a2, a3, a4 € R}.

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A: We plug x=0 and y=0

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A: Irreducible polynomial and it's order

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A:

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Q: Let f: IR → R and g: R → R with f(x) = 5x – 7 and g(x) = x2 – 4x + 4 Find: a) Domain of . g(D(f.g))…

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Q: The function

A: Given, the function f : ℂ→ℂ defined by                   f(z)=ez+e-zhas

Q: 5A Given the inner product space of polynomials with real coefficients, degree ≤n (P₁(R),+,., ° ),…

A: The angle between the vectors in inner product space:

Q: his is some (a) Use this

A: Given  (a) y" + 3y' + 2y = f(t), y(0)=y'(O)=0. (b)f(t) = e3t and f(t) = t.

Q: Find real numbers x, y and z such that (x + 4)i + (y – 5)j + (z – 1)k = 0

A: To Determine :-  The real numbers x , y and z  such that   x + 4  i +  y -5  j +  z - 1  k =0

Q: (d) Show that T is bounded and find ||T||. (e) Find the adjoint T* of T.

A: We will use the vector/matrix notations to visualise things easily.

Q: Let p be a prime and let q = p". Let f(x) = x³+ax+b € Fq[x] be

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Q: Find the norm of f(x) = √√1-x² on the interval [- 1,1]. 2√3 3 O O O W|N H

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Q: Suppose that β is a zero of f (x) =x4 + x + 1 in some extensionfield E of Z2. Write f (x) as a…

A: Given that β is the zero of fx=x4+x+1 in some extension field E of Z2. We have to write fx as a…

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: Let x0, x1, ..., Xn be distinct points on the interval [a, b], where Xo <x₂ < x₂ <--- < x₂ and let f…

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Q: p(1) p(2) p(3) mation, and find ker (T). Recall that P3 is the space of all polynomials of degree at…

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Q: Determine if the following polynomials are irreducible or not k(1, y) = 1® – y° € Q!r, y]

A: Given:- k(x ,y)=x6-x7 ∈ Q x,y

Q: 2c

A: The answer has been provided.Explanation:

Q: Find the linearization L(x) of f(x) 5x2 5x + 5 at x х — 2. a) L(x) 15x 15 O b) L(x) 25x + 15 c) L(x)…

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Q: 4. Find a polynomial p(x) of degree at most 2, which is not in the span of P₁(x) = x² - 2x +1,…

A: p1x=x2-2x+1p2x=x-1p3x=x2-1

Q: p(1) Р(2) p(3) mation, and find ker (T). Recall that P3 is the space of all polynomials of degree at…

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Q: 4. Define the operators T and U on the space P(R) of all polynomials in variable x as follows:…

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Q: 3. Let x0, x₁,...,xn be distinct points on the interval [a, b], where Xo <x₁ < x₂ <... < xn and let…

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Q: If possible make an explicit bijection from [0, 1] to [-2, -∞. If no possible, Why? If…

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Q: (d) Let ø(x) = x² + x, x > 0. If defines an even extension of o to the entire real line, then for a…

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Q: prove V injective if x1 ‡xn → v(x1) = v(xn)

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Q: . Let a := V1+ v3. Find the minimal polynomial of a over Q.

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Q: 3. Determine if the follwing function are analytic or not a. f(z) = (z*+1)³ – 3z* where z* is the…

A: We need to check whether the given function is analytic or not. a) given f(z)=(z*+1)3-3z*, where z*…

Q: Given the inner product space of polynomials with real coefficients, degree <n (P₁(R),+,., o),…

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Q: z? f(2) = 1-z* 4

A: We use the fact that the 1/(1-z)= 1+z+z2+z3+z4+......

Q: Let T: P₂- R² be defined by +T (PO) = [PⓇ]. (a) Find a polynomial p(t) in P₂ that spans the kernel…

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