14. The linear transformation L defined byL(p(x)) p(x) p(0)maps P3 into P2. Find the matrix representation ofL with respect to the ordered bases [x2, x, 1] and[2, 1 x. For each of the following vectors p(x)in P3, find the coordinates of L (p(x)) with respectto the ordered basis [2, 1 - x]:(а) х? + 2х — 3(b)(d) 4x22x(с) Зх

The linear transformation is defined by L: P3→P2 such that L(p(x)) = p’(x) + p (0), also given that…

Answered: 14. The linear transformation L defined…

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.
7. Let T: R³ R³ be defined by (a) (b) (E) x1 + x2 + x3 = X1 X2 X3 [X1X2 X3] Find the matrix of linear transformation for T. T Find all vectors 7 € R³ such that T(x) = x.
y+2 (O)-() x + (-)-( (b) Consider the linear map T: R³ R³ such that Ty i. State the matrix of T in standard basis B of R³. ii. Determine the matrix of T in the basis C = You may assume without proof that C forms a basis. 2 2 of R³.
-9 V₁ = --[-] and X₂ such that T(x) is Ax for each x. Let x = A = X₁ -2 [3] and let T: R² R² be a linear transformation that maps x into x₁v₁ +X₂V₂. Find a matrix A -8 and V₂ =
Solve ODES: (2) Solve (x² - y2-yz)p+ (x²-y²-zx)q = z(x-y). (3) Solve (x² + y² + y2)p+ (x² + y² -xz)q = z(x + y). (4) Solve (x² - y²-22)p+ 2xyq = 2xz. (5) Solve x(z² - y²)p+ y(x²-2²)q=z(y²-x²).
a) Verify the given sequence B is a basis ; and b) compute the images of the standard basis vectors with respect to the given linear transformation. 15. fi = 1+2x – 2x2, f2 = 2 + 3x – 4x2, f3 = 1 + 4x – x², B = (f1, f2, f3). - 1 T(fi) = (; 10)- 4 T(fi) = (; ).T ,T(f2) : 3 8
8-3) show that T is a linear transformation by finding a matrix that implements the mapping. Note that x1, x2,... are not vectors but are entries in vectors a) T(x1, x2) = (2x2-3x1, x1-4x2, 0, x2) b) T(x1, x2, x3, x4) = 2x1 + 3x - 4x4 (T:R* → R)
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Determine the criterion of the transformation T, where T: R² R² is defined on the basis B = {(6,1,5,1)} With T (6,1)= (6, −4) and T (5,1)= (-2,8)
A small country is comprised of four states, A, B, C, and D. The population of each state, in thousands, is given in the following table. State Total A B C D Population (in thousands) 205 465 845 1085 2600 According to the country's constitution, the congress will have 65 seats, divided among the four states according to their respective populations. a) Find the standard divisor, in thousands. How many people are there for each seat in congress? standard divisor: number of people for each seat: (Type an integer in each box.) b) Find each state's standard quota. (Two of them have been done for you.) c) Find each state's lower quota and upper quota. State Lower Quota Upper Quota A (Type an integer in each box.) State A B 205 465 1085 Population (in thousands) Standard quota 5.13 27.13 (Type an integer or a decimal rounded to two decimal places, if necessary, in each box.) State Lower Quota Upper Quota B (Type an integer in each box.) C D State Lower Quota Upper Quota C (Type an integer…
7. (a) Consider the ordered basis ß = [1,1 +x, 1 + x + x²] for V = P₂ and F = R. Suppose is in P2 and you know these coordinates: [] = (1, 2, 3). Find the vector 7. (b) Let A be a 2 x 2 matrix and a be a real number. Prove that det (a A)=a² det(A)

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