9. Let P3(C) be a vector space over C and let D: P3(C) → P3(C) be the+bt+ct²+dt³.differential operator defined by D(p(t)) = dp/dt. Let p(t) = aFind the matrix of D relative to the basis {1, t, t², ³).10. Consider the basis S = {(1,0), (1, 1)} of R². Let L: R²2 R2 be definedby L(1,0) = (6, 4) and L(1, 1) = (1,5). Find the matrix representation of Lwith respect to the basis S.

Since you have posted a multiple question according to guildlines I will solve first (Q9)question…

Answered: 9. Let P3(C) be a vector space over C…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4. Let W be the xy-plane in R³. (a) Find an orthonormal basis ū₁, 2 for W. (b) Construct the matrix Q = [ ₁ ü₂ ] and calculate the matrix QQT. (c) Consider the linear transformation T: R³ → R³ that projects each vector orthogonally onto the xy-plane. Note that T() = QQT. Use the matrix to calculate T(₁), T(2), and T(ē3). (d) Explain how your answers make sense geometrically.
Find the matrix A' for T relative to the basis B'. T: R² A' = X A' = 3 11/3 R², T(x, y) = (x - y, y - 2x), B' = {(1, -2), (0, 3)} -3 48 -16 -4 Find the matrix A' for T relative to the basis B'. ↓ 1 T: R² → R², T(x, y) = (-7x + y, 7x - y), B' = {(1, −1), (-1,5)} -72 24
Let f(u, v) = (u - v, 1, u + v) and g(x, y, z) = xyz. Find the entry a12 (i.e. the entry in the first row and second column) of the derivative matrices Df(u, v), Dg(x, y, z) and D(gof)(0, 1). (a) a 12 of Df(u, v) is (b) a12 of Dg(x, y, z) is (c) a12 of D(gof)(0, 1) is (d) Select the correct answer about D(g. f)(u, v) D(gof)(u, v) is a 1 x 2 matrix. D(gof)(u, v) is a 2 x 2 matrix. D(gof)(u, v) is a 3 x 2 matrix. D(gof)(u, v) is a 2 x 3 matrix. D(gof)(u, v) is a real-valued function of u and v. (e) Select the correct answer about D(f g)(x, y, z) OD(fog)(x, y, z) is not defined. D(fog)(x, y, z) is a real-valued function of x and y. D(fog)(x, y, z) is a 3 x 2 matrix. D(fog)(x, y, z) is a 2 x 2 matrix. D(fog)(x, y, z) is a 2 x 1 matrix.
Let f(u, v) = (u - v, 1,u + v) and g(x, y, z) = xyz. Find the entry a12(i.e. the entry in the first row and second column) of the derivative matrices Df(u, v), Dg(x, y) and D(g. f)(0, 1). (a) a12 of Df(u, v) is (b) a12 of Dg(x, y, z) is (c) a12 of D(g)(0, 1) is (d) Select the correct answer about D(g. f)(u, v) OD(g. f)(u, v) is a real-valued function of u and v. D(g. f)(u, v) is a 2 x 2 matrix. D(g. f)(u, v) is a 2 x 3 matrix. D(g f)(u, v) is a 3 x 2 matrix. 0 D(g f)(u, v) is a 1 x 2 matrix. (e) Select the correct answer about D(f g)(x, y, z) OD(fog)(x, y, z) is a real-valued function of x and y. D(fog)(x, y, z) is a 2 x 1 matrix. D(f g)(x, y, z) is not defined. 0 D(f g)(x, y, z) is a 3 x 2 matrix. D(f g)(x, y, z) is a 2 x 2 matrix.
Find the standard matrices A and A' for T = T₂ o T₁ and T' = T₁ 0 T₂. T₁: R² → R², T₁(x, y) = (x - 5y, 2x + 3y) T₂: R² R², T₂(x, y) = (0, x) A = Juma A' = SHE
9. Let and B 3 = {[B]-[-2]} = B' ={[2¹].[6]} be two ordered bases for R². Find the change of basis matrix [I]B'.
5. Find the standard matrix A and A' for T = T2° T1 and T' = T1° T2, where T1:R2 → R3, T1(x,y) = (x,x+y,y) and T2:R3 → R², T2(x,y,z) = (0,y). Use standard basis vectors to derive your re- sults.
Consider the function T : R → R², defined by T(r, y, 2) = (x – 2y + 2,1 – 2) (a) Write the matrix for T. (b) For the vectors u = (2,3, – 1), v = (-1,3,5) € R*, verify that T(3u + 4v) = 3T(u) + 4T(v). (c) For the unit vectors i = (1,0, 0), j = (0,1,0), k = (0,0, 1), write the matrix [T] = [T(i) T(j) T(k)]. (i.e. write the matrix [T] whose columns are the vectors T(i), T(j), T(k)) %3D
Let c₁ (t) = esti + 3 sin(t)j + t7k and c₂(t) = e−³¹i + 2 cos(t)j – 8t¹k. (Enter your solution as a single vector using the vector form (*,*,*). Use symbolic notation and fractions where needed.) -[c₁ (t) + c₂ (t)] = dt
Find the general solution by Variation of Parameters. y''+y=3cscx
Let A(t) = f(t)i + f₂(1) and B(t) = g₁(t)i + g₂(t)},t = [0, 1], where f₁f2, 81, 82 are continuous functions. If A(t) and B(t) are non-zero vectors for all and 7(0) = 2î +3ĵ, A(1) = 6î +2), B(0)=31 +23 and 6i- A B(1) = 27 +6, then show that A(1) and B(t) are parallel for some t.

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9. Let P3(C) be a vector space over C and let D: P3(C)→ P3(C) be the differential operator defined by D(p(t)) = dp/dt. Let p(t) = a+bt+ct²+dt³. Find the matrix of D relative to the basis {1, t, t², 3).