2. Let T: P₂ (R) → P₂(R) be defined by T(p)(x) = p(0) + p(1)(x+x²). (a) Prove that T is diagonalizable. (b) Find a basis D for P2 (R) such that [T]p is a diagonal matrix. Note that the basis D should consists of polynomials. (c) Compute [T]D (which should be a diagonal matrix).

Q: 4 Let A = and define T : R? → R? by T(x) = Ax. Find a -2 -1 basis B such that the matrix [T]s is…

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Q: Problem 2 : Let T : R3[r]→ R3[x] be the linear operator defined as T(p(x)) = (x + 6)p (x) + p"(x),…

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Q: Q01. Let S = {1, x, x², x³} denote the standard basis of P3(F) and let C = {To(x), T₁(x), T2(x),…

A: The notation sPc use for the transition matrix from basis C to basis S.

Q: 2. Consider the linear transformation T: P2(R) → P2(R), defined by T(f(x)) : f'(x) + 5f(x). Find the…

A: The linear transformation,   T:P2ℝ→P2ℝTfx=f'x+5fx

Q: For the linear transformation R[x]≤ 3 --->  R[x]≤ 3 defined by T(p(x))=p(x-1), Find the matrix…

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Q: 2. Let V P3(R) and F = R. If W1 W2 = {p(x) E P3(R) | p(1) = 0}, then {p(x) € P3(R) | p(0) = 0} and…

A: Result: If V is finite dimensional vector space and W is a subspace of V then , dim(W) = dim(V) -…

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: Let T: P3 (R) → R2 be defined by T(p) = (p"(0), p'(0)).

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Q: Consider the subspace U of R₁[X] given by U = {a+bx+cX² +dX³ +eX4:a+2c= 0 and b-3c=0} Find a basis…

A: The given subspace,.The aim is to find the basis of the subspace and the dimension of the subspace.

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: Let T: R² → R² be defined by T x1 = x1 x2 Let u = [2] and C = V1 = , V₂ = [A]} x1 Find T(u), the…

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Q: Consider the ordered basis B = {(1,-1,0), (1, 1, 1), (0, 1, 1)} of R³. the coordinate vector v =…

A: Consider the ordered basis B=1,-1,0, 1,1,1, 0,1,1 of ℝ3.   To Give: The coordinate vector v=-2,-1,0…

Q: (d) Let V = R³, W = R² and define T : V → W by T: (x1, x2, x3)(x₁ - x2, X2 - x3). Find the kernel of…

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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: -0 V₁ = V2 = [f] = = 1 7 1 -3 1 "-(⠀) -1 -3 2 -1 -32, Find the image vectors f (v₁), f (v₂), f (√3)…

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Q: the matrix of T relative to the basis B= {(1, 1), (- 1, 0)} and verify that [T], [a], = [T(a)]B Va e…

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Q: Let P2 denote the vector space of all one-variable polynomials of degree at most 2. Let B be the…

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Q: 4. Let W = {p(x) € P3(R) : p(-x) = -p(x)}. W is a subspace of P3(R) (you do not need to show this).…

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Q: Consider the basis S- (V₁, V2, V3) for R³, where v₁ -(1, 1, 1), v₂-(1, 1, 0), and v3 - (1,0,0), and…

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Q: 2. Let T : P2 –→ P2 be defined by T(a + bx + cx²) = a – 26 + (2b – 4c)x + 3 cx?. Find a basis C for…

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Q: (a) Prove that the set ẞ = {p0(x), p1 (x), ... , Pn(x)} of Lagrange polynomials is linearly…

A: Given: ẞ = {p0(x), p1 (x), ... , Pn(x)} To Prove" (a). Prove that the set ẞ of Lagrange polynomials…

Q: Problem 2 : Let T: R3[r] → R3[r] be the linear operator defined as T(p(x)) = (x+ 6)p' (x) + p"(x),…

A: Solution

Q: 3: Consider the linear map T:R³ → P2 given by a T( ) = (a – 26) + (3a + 46)x + (-3a + 6b)x². a) Find…

A: Let us consider the linear transformation, T: V→W Kernel of T is obtained as Ker(T)=v∈V|T(v)=0.…

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: Let T: R³ → R³ be a linear operator and B = [v₁, V2, V3] a basis for R³. Suppose 0 T(v₁) =_=A T(v₂):…

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Q: Let p(t) = f be non-zero constant polynomial, let g(t) = dt+e be a linear polynomial (d‡0), and let…

A: Given that, p(t)=f;                   q(t)=dt+e;                   r(t)=at2+bt+c; Now, we can write…

Q: For the linear map given by the matrix A below, you may use without calculation that chT (x) = (x −…

A: Given That : A characteristic polynomial chTx=x-13 and matrix A=432-4-3-3112. To Find : i) Minimal…

Q: 4) Let : R³ → R[t]<2 be the linear map that sends 4(e₁)=t+t² 4(e₂) = 1-t 4(e3) = t Write the matrix…

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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: Consider the map T : P2 H→ P2 given by T(p(x)) = p" (x) + p(x) Find [T] in terms of the basis ß = {1…

A: Matrix represetation using the basis

Q: Define T: R² →R² by T(x) = Ax, where A is the matrix defined below. Find a basis B for R2 with the…

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