s) Let FR" →→R" and be a linear transformation. Suppose the matrixiepresentation of F with respect to the standard basis B := {e₁, en} CR" is ann x n matrix A. Show that the following linear transformationF(3): R → R", F³) (v):= F(F(F(v)))has matrix representation with respect to the basis B given by the matrix A³. Justifyyour answer.

Answered: Image /qna-images/answer/b6e703e3-89a1-4e4b-b5b0-bbe774d277c4.jpg

Answered: s) Let FR" → R" and be a 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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Let S be a linear transformation from R2 to R2 with associated matrix 11. Let T be a linear transformation from R2 to R2 with associated matrix -F A = C= B = 2 Determine the matrix C of the composition To S. -3 2] 0 2
Let T and T2 two linear transformations from R? to R², defined by 7(;) - ) ()-() 3x - 2y (;) (} x+ y T1 T2 y - 3x for all (x, y)" e R?. Find the standard matrix representations for each of the following composite functions, and their inverse transformation, if they exist, by using the algorithm for matrix inversion in Lemma 3, Topic 14. )T followed by T2. (i) T2 followed by an anticlockwise rotation by angle 5 in R?. (ii) A reflection in the line y = 2x followed by a projection onto the line y = -3x in R2.
The matrices A₁ = A3 [] = ^--^- A₂ = [] =[] A4 form a basis for the linear space V = R 2x2. Write the matrix of the linear transformation T: R 2x2R 2x2 such that T(A) = 3A + 10AT relative to this basis.
Suppose M is a linear transformation taking vectors in R² to vectors in R2, where M(1,0)=(2,-4) M(0, 1) = (-6,6) M(2,9) = (
help
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)
'
C: Let T be the linear operator associated in the standard basis B of F2 with the matrix 2 - [44] - 1 Let B' be the basis ((2, -1), (2, 1)) M = b) Using eachof these three matrices find the dimension of the range of T (you must check that the dimension is the same with all the matrices)
Please explain
Suppose T: M2,2→→P2 is a linear transformation whose action on a basis for M2,2 is as follows: [18]-- Describe the action of T on a general matrix, using x as the variable for the polynomial and a, b, c, and d as constants. Use the character to indicate an exponent, e.g. ax^2-bx+c. = -2x+5 T = 0 1 1 -2 -2 = -x+4 T -4 -2 = 2x² +4x-26 T 00 = -2x²+2x+10
is a) and b) linear transformations. If yes, write down the matrix of the transformation
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