9. Find an orthonormal basis for P[r]3 with a basis 1, 2, 2² if the inner product on P[r]3 is defined by (f(x), g(x)) = [ f(x)g(x)dx

Q: (a) Let Pn(R) denote the vector space of polynomials of degree at most n, where addition is the…

A: Given :  T : P3(ℝ) → P2(ℝ)  defined by                  T(p(x)) = 3p''(x)-2p'(x) To find :(a) We…

Q: 6. Suppose V is a complex vector space with dimension n and T E L(V) is invertible. Let p and q be…

A: In the question it is asked to prove p(0)q(z)=znp(1z).

Q: Let V = P2(R), give V the inner product defined by (f,g) = Lo s(1)g(t) dt, and let W = span {1 – x}.…

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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: Consider P2 (R) together with inner product: Find an orthogonal basis for P2 (R). (p(z),9(2) = Г'р…

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Q: 6. Let T: P₂ → P₂ be the linear operator defined as T (p(x)) = p(3x), and let B = {1, x,x²} be the…

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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: Let P₂(R) be the vector space of all real polynomials of at most degrees 2 with the inner product…

A: Given that

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: 10. Let P denote the vector space of polynomials of degree at most k. Let T: P3 → P2 be the function…

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Q: Find a basis {p(x), g(x)} for the vector space {f(x) = P₂[x] | f'(3) = f(1)} where P₂ [x] is the…

A: Here we have to find the p(x) and q(x) that are basis of given vector space.

Q: Consider a function space in (-∞, ∞) with orthonormal basis {n}. Show that any function ƒ(x)…

A: Answer:-

Q: Let L: R2 R2 be defined by L 4*]-[***] represents the coordinates of v in the standard ordered…

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Q: Find a basis {p(x), q(x)} for the vector space {ƒ(x) = P₂(x) | ƒ'(8) = ƒ(1)} where P₂(x) is the…

A: For any vector f(x) ∈ P2(x) , there exist scalars a, b, and c such that         f(x) = a + bx + cx2…

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: Find a basis {p(x), q(x)} for the vector space {f(x) = P₂[x] | ƒ'(-3) = f(1)} where P₂[x] is the…

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Q: In the vector space of real functions on [-, π], let the inner product be defined by ㅠ (f,g) = f( […

A: Using the Gram-Schmidt orthogonalisation prosess to find an orthogonal basis.

Q: Let V be a finite-dimensional inner product space. A linear operator U on V is called a partial…

A: A partial isometry is a linear operator on a finite-dimensional inner product space that preserves…

Q: Let T: R² R² be defined by T Let u = T(u): ¹([^])=[^²=²] = X2 H | amnet C = { v₁ = [!] > v₂ = [2]}…

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Q: 7. Let V be an n-dimensional vector space over the field F and let B = {V₁, V2, ..., Vn} be an…

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Q: Find a basis {p(x), g(x)} for the vector space {f(x) = P₂(x) | ƒ'(-3) = f(1)} where P₂(x) is the…

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Q: State with reasons whether each of the following maps T : P2(R) → P2(R) is linear. For those maps…

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Q: (a) Show that the list {p o(x), p 1(x), p 2(x), ...,p n(x),...,} where po(x) = 1, p1(x) = 1+x, P;(x)…

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Q: Let 1 0 1 2 0 -1 -1 1 0 0 1 1 0 1 1 and let T: R6 →→ R5 be defined by T(x) = At. (a) Find a basis…

A: Given matrix is  and (a) Obviously  basis for  is same as basis for column space of A .         Now…

Q: Let S = {1,2,3,4} and U = {ƒ € Fun(S) | ƒ(2) − 3ƒ(4) = 0, ƒ(3) = 0} Find a basis of U. Basis(U) = {}…

A: Given  S = 1, 2, 3, 4  and  U =  f ∈ Fun(S) / f2 - 3f4 = 0 , f3 = 0                      Let  f ∈ U…

Q: On P(R), consider the inner product given by 1 (p · q) = ["'p(x)q(x) dx. Apply the Gram-Schmidt…

A: Given inner product is Let Using Gram Schmidt Process to make  orthogonal. Let Using Gram Schmidt…

Q: Find a basis {p(x), g(x)} for the vector space {f(x) = P₂(1) | f'(-2) = f(1)} where P₂(x) is the…

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