Let L: R4 → Rª be the linear transformation represented by the matrix3 0 0 04 10 00 0 2 10 0 0 2with respect to the ordered basis B = {v1, v2, V3, V4}.(a) Let v E R“ such that v = vị + 2v½. Find L(v).%3D(b) Is v an eigenvector of L? Justify your answer.

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Answered: Let L: R' → R“ be the linear…

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 4: R³ → R[t]<2 be the linear map that sends Y(e₁) = t + t² 4(e₂) = 1-t 4(e3) = t Write the matrix of the complexified map yc : C³ → C[t]<2 with respect to the complexified standard basis of both vector spaces.
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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 smallest integer value for K so that the linear ,3 operator T:R°→R° given by T(xy,z) = (x + y +z,kx +y – z,x + y +z) be degonalizable. Then, for the value of K found, determine a diagonal matrix similar to the matrix of the operator T on the canotic basis of the R
Suppose A is the matrix for T: R3 R³ relative to the standard basis. Find the diagonal matrix A' for T relative to 1 A'= 2 0 0 1 -1 0 0 0 -1 B'= {(1, 1, 0), (2, 1, 0), (0, 0, 1)} A = -2 0 0 0 1 0 11 F 0 0 1
D: P2→ P2 (Dthe differential operator so that D(ax²+bx+c) = 2ax +b); a the standard basis for P2; ß the basis consisting of x2+1, x+1, 2x²+1; ở = x² + 2x – 2. a) Find [T]% where a is the standard basis for P2 b) Find the change of basis matrix from a to B. c) Find the change of basis matrix from B to a.
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.
Suppose that T:R²→ R³ is a linear transformation and T(1,-3)=(4,–3, 1) and T(1,4)=(1,3, 1) and let B={(1,–3),(1,4)}. a) Find the matrix representation of Tin terms of B and S3 [T] where S3 is the standard basis for R'. b) Find the change of basis matrix [C, to convert from the standard basis in R? to B. c) Find [T] = [TCE, ->
Suppose T is a linear transformation on R² (i.e. T(x, y) = (ax + by, cx + dy) where a, b, c, d are real numbers and, interpreting T as a matrix, det(T) # 0. Let P be the parallelogram which is the image of the square D = = [0, 1] × [0, 1] (i.e. T(D) =P). Then the area of P is given by O lad - bcl ad bc abcd 1 |ad - bc| O labcd|
Consider the following matrices: √/3 Q 8 M' = – 051010 100 10+ 9 (a) Find the component q in the coordinate transformation matrix Q such that the matrix is orthogonal and associated with rotation. q= = (b) Let Q transform the orthonormal basis set B to B'. Determine the components of matrix M' in the new orthonormal basis set B'. (c) Calculate the three invariants of M. I₁ I₂ = M = (d) Calculate the three invariants of M'. I I! = [2 −1 6 7 -5 6 1 -1 4 I3 = =

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Let L: R' → R“ be the linear transformation represented by the matrix 3 0 0 0 4 1 0 0 0 0 2 1 |0 0 0 2 with respect to the ordered basis B = {v1, v2, V3, Va}. (a) Let v E Rª such that v = vị + 2v2. Find L(v). (b) Is v an eigenvector of L? Justify your answer.