1) The linear transformation L defined by L(p(x)) = p'(x) + p (0) maps P3 into P2. (a) Find the matrix representation of L with respect to the ordered bases {1,x,x2} and {1, 1-x}. (b) For the vector, p(x) = 2x + x - 2 (i) find the coordinates of L(p(x)) with respect to the ordered basis {1, 1-x}, using the matrix you found in (a). Remember to use the coordinate vector of p(x) with respect to the basis {1, x, x2}. (ii) Show that they are the weights that work by writing the linear combination with the basis elements and comparing the resulting polynomial to L(p(x)).

1) The linear transformation L defined by L(p(x)) = p'(x) + p (0) maps P3 into P2. (a) Find the matrix representation of L with respect to the ordered bases {1,x,x2} and {1, 1-x}. (b) For the vector, p(x) = 2x + x - 2 (i) find the coordinates of L(p(x)) with respect to the ordered basis {1, 1-x}, using the matrix you found in (a). Remember to use the coordinate vector of p(x) with respect to the basis {1, x, x2}. (ii) Show that they are the weights that work by writing the linear combination with the basis elements and comparing the resulting polynomial to L(p(x)).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: L is the left multiplication transformation defined by = Ax L₁(x) = A=[T] vector space and Suppose…

Q: Suppose the vectors u and v are a basis B for R^2 and T is a transformation from R^2 → R^2 that maps…

A: Hello. Since your question has multiple parts, we will solve the integration by first method for…

Q: The linear transformation_T:R³ OK² is represented. by the matrix [T]8= 1 20 where B= 012 and B' = 二…

Q: 4. (10 points) Let u₁ = , U₂ = - 12₁²-4₁ , V2 = on R2 whose matrix representation with respect to…

A: Given two bases of R2:

Q: L(p(x)) = - 6p' – 3p" maps P4 into P3. (a) Find the matrix representation of L with respect to the…

A: Given L:P4→P3 defined by Lpx=6p'−3p''. a We have to find the matrix representation of L with…

Q: Find vectors VI 8 in R such that S is the transition matrix from (v₁, ₂) to (₁.₂). (b) Let P, be the…

A: Note: " As per our guidelines we will solve the first question. If you want any specific question to…

Q: Show that any isometry of R^n which fixes the origin is linear, and hence must be of the form f(x) =…

A: Introduction A square matrix is said to be an orthogonal matrix if…

Q: Define a linear operator (transformation) on R2 by performing the following operations in order: (i)…

Q: 5) Let P₂(C) be the vector space of polynomials with complex coefficients of degree ≤ 2, equipped…

Q: IfT: R² R³ is a linear transformation such that -n the standard matrix of Tis 13 T([2])-[3] . and T…

Q: Suppose that T : R² → R² is a matrix transformation and that 7([}))-[-), "([;))-}] 7((:) (a) Find…

Q: For this actvity, find the matrix representation [7] for the linear transformation 7 : R³ → R²…

Q: The linear tranformation L defined by L(p(x)) = -13p' – 14p" maps P, into P3. (a) Find the matrix…

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…

Q: 1s) La L : A → R be the limsir operator Lip(z)) – p(0)r*+27/(c). Find the matrix A representing with…

Q: The linear tranformation L defined by L(p(x)) = -2p' +9p" maps P4 into P3. (a) Find the matrix…

Q: T is linear transformation given by: p(x) => x^2(p(1/x)) Compute the matrix of T with respect…

Q: Define T : P4(R) → P4(R) by T(p)= ((3x^2 −4x+9)p)'' (a) Show that T is a surjective linear map.…

A: It is given that T:P4(ℝ)→P4(ℝ) defined as T(p)=((3x2-4x+9)p)'' for p∈P4(ℝ)Basis for P4(ℝ) is {1, x,…

Q: Let k1, k2, ..., ks be positive integers and consider sx s block diagonal matrix J(A) given by Jk₁…

Q: Зи — v Let T : R? → R² be the linear map defined by T () = C) 3. with {() (} -{() (} and B' = basis…

Q: (a) Find a single matrix that defines a rotation of the plane through an angle of -(π/4) about the…

A: We have to find a single matrix that defines a rotation of the plane through an angle of -(π/4)…

Q: Harry fills up his Jeep with gasoline and notes that the odometer reading is 23,511.9 miles. The…

Question

1) The linear transformation L defined by L(p(x)) = p'(x) + p (0) maps P3 into P2.
(a) Find the matrix representation of L with respect to the ordered bases {1,x,x2}
and {1, 1-x}.
(b) For the vector, p(x) = 2x +x - 2
(i) find the coordinates of L(p(x)) with respect to the ordered basis {1, 1-x},
using the matrix you found in (a). Remember to use the coordinate vector of p(x)
with respect to the basis {1, x, x2}.
(ii) Show that they are the weights that work by writing the linear
combination with the basis elements and comparing the resulting polynomial to
L(p(x)).

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Complexity$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

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.
Jn is simply an nxn matrix with only ones inside.
Day traders typically buy and sell stocks (or other investment instruments) during the trading day and sell all investments by the end of the day. The following table shows the closing prices on September 22, 2015, of 12 stocks selected by your broker, Prudence Swift, as well as the change that day. Tech Stocks Close Change AAPL (Apple) $113.40 -1.81 ADBE (Adobe Systems) $84.66 1.34 EBAY (eBay) $25.61 -0.31 MSFT (Microsoft) $3.90 -0.21 S (Sprint) $4.40 0.02 WIFI (Boingo Wireless) $8.51 0.56 Non-Tech Stocks ANF (Abercrombie & Fitch) $21.81 -0.02 в (Воeing) $133.99 -2.03 F (Ford Motor Co.) $13.91 -0.40 GE (General Electric) $25.10 0.01 GIS (General Mills) $57.12 0.33 JNJ (Johnson & Johnson) $93.26 0.13 On the morning of September 22, 2015, Swift advised you to purchase a collection of three tech stocks and two non-tech stocks, all chosen at random from those listed in the table. You were to sell all the stocks at the end of the trading day. (a) How many possible collections are possible?…
2. Let A = be an orthogonal 2 × 2 matrix. cos(0) sin(0 (a) Explain why the vector can be written as for some value of 0. a Use your answer to part (a) to find all 2×2 orthogonal matrices A = |C (b) (Hint: What can the second column of A look like?) (c) det(A) = 1? What kind of linear transformation do they correspond to? ! Given your answer to part (b), what are the 2 × 2 orthogonal matrices with
Construct a homogeneous transformation matrix with the provided rotation and translations. R 7 Hint: Remember that a homogeoneous matrix looks like 0 1
Consider the linear transformation T: R^ →R" whose matrix A relative to the standard basis is given. [³ A = (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (1₁, 12) = 4,5 B₂ 3-2 (b) Find a basis for each of the corresponding eigenspaces. B₁ = 1,2 128 1,1 = A' = 1 6 (c) Find the matrix A' for T relative to the basis B', where B' is made up of the basis vectors found in part (b). X 5 5 1 2 ↓ 1
2 [H] a) Treating u, v', and w' as vectors, are the following inner products defined? (i) u. v' 6) (True/False) Let u = (ii) v'. u (iii) u - w' v' = [1 3 2], and w' = [5 -1 -1]. b) Treating u, v', and w' as matrices, are the following matrix products defined? (i) uv' (ii) v'u (iii) uw'
Consider the vector space V of all the real polynomials of degree less than or equal to three, p(x) = ax^3 + b x^2 + c x + d, and consider the linear transformation T: V ------> V given by the derivative, i.e., T( p ) = p' = 3ax^2 + 2 b x + c. Find the eigenvalues and eigenvectors of this transformation.