Let te R and let g₁(x) = 1, Prove that {g₁, 82, 83} is a basis of 8₂(x) = x+t, g3(x) = (x + 1)². P₂ = {ao + a₁x + a₂x² | ao, α₁, a₂ € R}.

Q: 6. (Basis and Dimension.) Say P3(R) = {f: R → R | f(x) = ax³ + bx²+cx+d for some a, b, c, d = R}…

A: The objective of the question is to determine which of the two collections of functions, W₁ and W₂,…

Q: Be S = {v1 = (1,2, 1), v2 = (2,9,0), v3 = (3, 3, 4)} a subset of R and T : R³ → R? T(v1) = (1,0),…

A: Given that  S=v1=1, 2, 1, v2=2, 9, 0, v3=3, 3, 4 be a subset of ℝ3 and     T : ℝ3→ℝ2      Tv1=(1,…

Q: Let T : R2[x] → R2[x] be given by T(a+bx+cx2 ) = a + b(x+ 1) + b(x + 1)2 . Let v = x + 1. Find a…

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Q: 3. Let p₁(x) = ½x(x − 1), p2(x) = −x² + 1 and p3(x) = ½x(x+1). - (a) Show that E = {P1, P2, P3} is a…

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Q: 59. In R4, given 2 nullspaces 1 2 00 1 V : |-1 0 -1 m 0- Find m such that dim(U n V) is max. Find…

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Q: the coordinates of (2, 3, 4, – 1) with B = {(1, 1, 1, 2), (1, –1, 0, 0), (0, 0, 1, 1), (0, 1, 0, Q}}…

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Q: Let B = {1, x, x², x*} be an ordered basis for P3. Find the coordinate vector of f(x) = 4x3 – 72² +…

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Q: 7. Prove that p, (t) = 1+t², p2(t) = t – 3t2, p3(t) = 1 +t – 3t, is a basis for P2.

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Q: Let f E P(R) be a polynomial of degree n. Let yo,..., Yn be the dual basis of 1, x, x²,..., x". Show…

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Q: b. Prove that the polynomials p₁(t)=1, p₂(t)=1-t, and p₂ (t)=1+t-t², form a basis for P₂.

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Q: { 1 + x + x², x + x², x²}. %3D for P2. Let p(x) = -1 +x + 2x2. Then [p(x)]B = %3D %3D

A: The given basis is B=1+x+x2,x+x2,x2 for p2. let p(x)=-1+x+2x2 We have to find,  p(x)B=?

Q: Let s = {1, 2, 3} andT: Fun(S) + R' be the transformation T) = (10), s) – 2/), 1() – 12) and…

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Q: Let B = {1+t, t – t2, 2t + t2} be a basis for P2. Find [-3+t – t°]B. %3D

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Q: {e₁,e2, e3} be the standard basis of R³ and let f,g : R³ → R³ be the linear map 1. Let E =…

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Q: D: P2> P2 (Dthe differential operator so that D(ax²+bx+c) = 2ax +b); a the standard basis for P2; B…

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Q: 3. Let T : R² → R² be defined by T = X Y -x+y x-y

A: 3. Given that  be defined by:

Q: Find the coordinate vector [p(x)]B of p(x) = x² + 4x + 4 with respect to the basis B = {x³ + x – 2,…

A: According to the given information, it is required to find the coordinate vector.

Q: 4. Let A = [1 1 1]. Find a basis of N(A) and C(A).

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Q: Let T be a linear operator on R³ such that the rational form of T is [T] B = 0-20 1 0 for some…

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Q: Select all that constitute a basis of solutions for the following by the Frobenius method. (x+2)2y"…

A: Find a general solution to the ODE below.This is eligible to use the Frobenius method for linear…

Q: 1. D: P₂ - → P₂ defined by D(a₁ + a₁ + a₂x²) = a₁ + 2a2x and the basis B = {x+1, x − 2, 2x²} of P₂.…

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Q: The coordinates of p(x)=5+3x+x² relative to the basis {1+x,2-x2,3+x+x²} is O (-1,1,2) none (0,1,1) O…

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Q: Consider the basis S - (V₁, V₂} for R², where v₁ - (1, 1) and v₂ - (1, 0), and let T:R² R² be the…

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Q: Select all that constitute a basis of solutions for the following by the Frobenius method x²y" + 6xy…

A: Given differential equation isx2y″+6xy′+(4x2+6)y=0Rewrite the equation asy″+6xy′+4x2+6x2y=0Here…

Q: Let T: P₁ → R² be defined by T (P) Let q(t) = 5t6t² +7t³. Find T(q). Find a basis for the kernel of…

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Q: Let V = Poly4(R) and let B = (1, x2 − 1, x, x^3 − 2x, x^4 − 3x^2 + 1). (a) Find a basis C of V such…

A: Let V = Poly4(R) and let B = {1, x2 − 1, x, x3 − 2x, x4 − 3x2 + 1}. Note that here V is a vector…

Q: Take L: V --> W, from P2 --> P2 and L(p) = p + p', with basis B = {1 + x + x2, 1 + x, 2x + x2} and…

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Q: 9. Find an orthonormal basis for P[r]3 with a basis 1, 2, 2² if the inner product on P[r]3 is…

A: We have to find the orthonormal basis for  with a basis The inner product is defined by

Q: If the temperature at which a certain compound melts is a random variable with mean value 115°C and…

A: We have to solve given Questions.

Q: Find a basis of solutions by the frobenius method. Xy'' + 2y' + xy = 0

A: To find a basis of solutions for the differential equation xy″+2y′+xy=0 using the Frobenius method,…

Q: Let B = {1,1+x,1+ x + x² }. (a) Check whether that the set B is a basis for P2. 2 (b) Let p(x) =…

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Q: 1 -1 -4 3 -4 0 A basis is {v1, v2} where Let A 5 V1 || - Find a basis for Nul(A)+. V2 C

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Q: Define a basis v1, v2 of R² by vị obtained by applying the Gram-Schmidt Procedure to v1, v2? 2 by…

A: Consider the provided question, A Basis of a vector space. Let V be a subspace of Rn for some n. A…

Q: II Coordinates 5. Fix some a € R, and define polynomials Po(X), P1(X), P2(X) = P(R) by Po(X) = 1,…

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Q: - Suppose that f(x) = x² + 1, g(x) = x² - x and h(x) = x − 1. Then the set of functions {f, g, h} is…

A: DETAILED SOLUTION IS PROVIDED BELOWExplanation:

Q: Show that B = {v1, v2, v3} is an orthogonal basis for R3: v1 = (3,−3,0), v2 = (2,2,−1), v3 =…

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Q: If {x, a} is a basis of solutions of a?y" + axy' + by = 0, then a = (A) -2 (в) 2 (C) -3 (D) 3 (E)…

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Q: Let V = P4(R). Find a basis of V containing the polynomials p1(x) = 1 + 2x-x 2+3x 3+2x 4 , p2(x) =…

A: The detailed solution of the problem is given in the next steps.