Prove that the dimension of Pn(R), the set of all polynomials of degree less than or equal to n, is n+1, where R is the set all real numbers.

Q: Using the 10 properties: (a) Show that the set of strictly positive reals: R+ = {a E R | a > 0},…

A: (a)                          Given     R+ =  a∈R / a > 0            The operators ⊕  and  *…

Q: If F is a field then F[x,y] is not a P.I.D

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Q: Prove that the map g : Z →→ N defined by | 2k, -(2k+1), k < 0 k 2 0 f (t) = is one to-one nd mans…

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Q: Exercise 47 (a) Show that there exists a unique polynomial po of degree < 2 such that po(-1) = 1,…

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Q: Proving Polynomial Identifies 1) Show that (a + b)2 – (a – b)² = 4ab is a polynomial identity.

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Q: Q1:Let R={1, 2, , 103, .. A={xEN:1<xs 3}=D{1,2,3}, B={xeN:8< xs 103%={8,9, 103}, C={xEN:1<x< 2}={1,…

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Q: A) Let f: Z→ N as follows: 2n – 1 for n > 0, - f(n) = -2n for n < 0. Show that f is a bijection.…

A: Bijective function

Q: (b) Consider f(x) = 2x³ + 3x² + 4 in Z[], and the evaluation homomorphism 2 [2] : Z₁ [2] → Zg. (i)…

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Q: Please see the attachment. Linear Algebra

A: Given matrix is

Q: W = {(x1,x2,x3,x4,x5) ∈ R5 : x1 = 3x3 and x2 = 7x4}

A: W = {(x1,x2,x3,x4,x5) ∈ R5 : x1 = 3x3 and x2 = 7x4}, Now given condition are X1 = 3X3  and X2 = 7X4.…

Q: Find the dimension of the space of polynomials p(x) E P3 such that p(7) = 0, p(5) = 0, p(3) = 0

A: Clearly the polynomial has 3 roots. So the degree of the polynomial = k = 3.

Q: We define an addition operation on V by (a + bv2) + (c+ dv2) := (a + c) + (b + d)/2, for a +bv/2, c+…

A: The given set is,V=a+b2:a,b∈ℚThe addition and scalar multiplication on V is as…

Q: Suppose V is a complex inner product space. Prove that every normal operator on V has a square root.…

A: To prove: Every normal operator on V has a square root.

Q: Chooes In legendre polynomials Porm the bassis of chebyshev Polynomials Te CX)= ®늦 (4L201 -Locu) +…

A: Solution

Q: (C).(?)) under the standard inner product on C².

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Q: The product of 2x³ + 3x² + 1 and 2x² + 4 in z5[x]/ is: 5x^2+ 5x +1 O None X^2+2 4x+1

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Q: spectra of TT* and T*T are real and cannot

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Q: Duplicating the method of proof of Theorem 10,show that P(A∩B∩C∩D) = P(A)·P(B|A)·P(C|A∩B)·P(D|A ∩ B…

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Q: 78. W = {(s + 4t, t, s, 2s — t): s and t are real numbers}

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Q: What is the coordinate vector for the polynomial p(x) = 2 – x² + x'relative to the standard basis…

A: our objective is to find the coordinate vector for the polynomial p(x).

Q: "Consider the relations R₁ R₂ R3 R4 R5 on A = {1, 2, 3, 4} defined by R₁ = {(1, 2), (2, 1)} R₂ =…

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Q: Let R be a relation on the integers Z such that (a,b) = R if and only if ab = 0.

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Q: Using the Rodriguez formula obtain the Legendre polynomials Pn(x) for n=0,1,2,3,4,5

A: We have to find the Legendre polynomial Pnx by using  Rodriguez formula for n=0, 1, 2, 3, 4, 5. The…

Q: 2. Let T be the normal operator on the finite-dimensional complex inner product space V. Then T is…

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Q: number theory

A: If you want any specific question to be answered, then please submit that question only or specify…

Q: Let f(x) = x³ + x + 4 and g(x) = 2x + 3 in Z, [x]. Then the remainder when dividing f by g is: O 4

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Q: (b) Consider f(x) = 2x³ + 3x² + 4 in Zs [], and the evaluation homomorphism 2 [2]: Zs [2] → Zg. (i)…

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Q: Determine the multiplicity of the zeros of the following polynomials. 1. f(x) = (x - 2) 2. f(x) = (x…

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Q: Prove that if {v_1,...,v_n} spans V, then so does {v_1 − v_2, v_2 − v_3,...,v_(n−1) − v_n, v_n}.

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Q: obtained from f(x) by reducing all the coefficients of f(x) mo Assume that deg (f (x)) = deg(f(x)),…

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Q: Compute the product using the methods below. If a product is undefined, explain why. a. The…

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Q: Show that u = √3+ √2i is algebraic over Q and type its minimal polynomial p(x) = irr(u, Q) below.

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

Q: Show that H = 1² (1) is a a hilbert space

A: A Hilbert space is a a vector space which is equipped with the inner product that defines a distance…

Q: Let P2 denote the vector space of all polynomials in the variable x of degree less than or equal to…

A: Dear Student  As per our guidelines we can answer only first question.  So kindly resubmit the…

Q: The figure shows a transformation of y=1/x a) find a possible formula for the graph b) write the…

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Q: Given B = v1 {(0, 1, 1, 1)}, v2 = {2, 1, –1, –1} , v3 {(1,4, –1,2)} , v4{(6,9,4, 2)} B' = wi {(0,8,…

A: (b) T:R4→R3 is such that matrix Ais the transformation from B to B'. A=3-2101621-3071 Row reducing…