II Coordinates 5. Fix some a € R, and define polynomials Po(X), P1(X), P2(X) = P(R) by Po(X) = 1, p₁(X)=X+a, p₂(X) = (X + a)² (a) Prove that B = {po(X), P₁(X), p2(X)} is a basis of P₂ (R), the space of all polynomials of degree ≤ 2 with real coefficients. (b) For any f(X) = co + c₁X + c₂X² = P₂ (R), compute the coordinates of f with respect to the ordered basis B from part (a).

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: → R? and L(x) = (2x1 – x2, x2 +x3). Determine the Find the kernel of L, ker(L) where L: R3 dimension…

A: We have a linear transformation L:R3→R2  and Lx=2x1-x2,x2+x3. Now, kernel of a linear transformation…

Q: 2. Which of the following is a basis for the space of polynomials of degree < 2 over R? (Check all…

A:

Q: Which of the following sets of polynomials form a basis for P₂? (i) P₁ = 1 + 4x + 6x², P2 3+x = (ii)…

A:

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…

A:

Q: the basis of P2 cons tive to B.

A: Introduction: For some n, consider V to be a subspace of Rn. A basis for V is a collection of…

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

A:

Q: Let V be the set of all pairs (x,y) of real numbers together with the following operations: (x1,y1)…

A: Given  be the set of all pairs  of real numbers together with the following operations:

Q: Let V be the vector space of functions spanned by Nπ B = {b₁ = 5 sin x + 7 cos x, b₂ = 3 sin x + 9…

A:

Q: Show that S = {t+1,t-1,(t-1)2} be a basis of P2, the vector space of all second-degree real…

A:

Q: Let p1(t) = t and p2(t) = t + 1 be polynomials in P3. (a) Show that p1(t) and p2(t)} are linearly…

A: Given: p1t=tp2t=t+1 (a) We have to show that p1t and p2t is linearly independent. (b) We have to…

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: Let p₁ (t) = 7+t², P₂ (t)=t-2t², P3(t) = 2+t-4t². Complete parts (a) and (b) below. a. Use…

A: The polynomials p1t, p2t and p3t belongs to the vector space P2 because P2 contains all the…

Q: be the vector space of all polynomials of degree 3 or less in the variable z. Let PI(T) 2+x+1²,…

A: In this question, we check the given set forms a basis or not. Then we use elementary transformation…

Q: Find a basis {p(x), q(x)} for the vector space {f(x) e P2[x] | f' (7) = f(1)} where P2[x] is the…

A:

Q: Consider the following ordered basis for P₂ (R): B = {x² + 2x + 1, x + 1, 1} (a) The quadratic…

A:

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…

A:

Q: How do I find a basis for {(2c, c, b)} | b, c ∈ R}? I tried writing it as a linear combination and I…

A:

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: (a) (b) (c) Let W be the subpace of M2x2 (R) b W = { [20 + $ 30+d]:akedER} a, b, c, Find a basis for…

A: .

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…

A:

Q: hich of the following sets of polynomials form a basis for P2? 1, p2 1- 5х, рз || (i) p1 х, р4 (i)…

A:

Q: Define the linear transformation T : P2 → P2 according to T (p(x)) = p(2x − 5). (a) Compute the…

A: Given T(p(x)) = p(2x − 5)

Q: How can I find [L] from basis B to basis C, if basis B = {1 + x + x^2, 1 + x, 2x + x^2} and basis C…

A:

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) =…

A:

Q: The space P2₂ represents all 2nd degree or less polynomials. A polynomial such as p(x) = 9+8z +…

A:

Q: 4. Finding roots of polynomials can be hard. Finding polynomials with given roots can be done with…

A:

Q: Given that f is a polynomial of degree n in Pn, show that { f, f', f'', . . . , f(n)} is a basis for…

A:

Q: (a) Prove that for every positive integer n, one can find n + 1 linearly independent vectors in…

A: (a) To prove that for every positive integer , one can find linearly independent vectors in…

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…

A:

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)…

A:

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: T: P2 P₂ defined by T(a + bx + cx2) = a + b(x + 8) + b(x + 8)² Find the following images of the…

A: