5. Let B= (b₁,b2), where b₁ = (1,3), b2 = (-3,1).(a) Specify the transition matrices Ps,B and PB,s(b) For T the linear transformation corresponding to a reflection in the line y = 3r, specify[TB.8.

Introduction The transition matrix is the matrix formed by writting the one basis elements into…

Answered: 5. Let B= (b₁,b2), where b₁ = (1,3), b2…

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 B = {(1, 3), (-2,-2)} and B' = {(-12, 0), (-4,4)} be bases for R2, and let 3 2 40 be the matrix for T: R2 R2 relative to B. 56°F A = (a) Find the transition matrix P from B' to B. P = (b) Use the matrices P and A to find [v] and [T(v)]B, where [v]B = [-5 4]T. [V] B [T(v)]B = (c) Find P-1 and A' (the matrix for T relative to B'). p-1 = 4
Suppose T : R^2 → R^2 is a linear transformation that does not alter e_i andwhich maps e_2 into e_2 + 2e_1, where e_i = (1,0), e_2 = (0,1). Find the transformation matrix T.Justify your answer.
4a). The transformation defined by the matrix A = [14 5 stretches images in R² in the directions y = x and y = -x. Figure out the factor by which anything in the y = x direction is stretched and the factor by which anything in the y = -x direction is stretched. 4b). The transformation defined by the matrix B = -82 -55 13 3] stretches images in R2 in one direction by a factor of 3 and some other direction by a factor of 2. Figure out what direction gets stretched by a factor of 3 and what direction gets stretched by a factor of 2. 4c). The transformation defined by the matrix C = stretches images in R2 in two directions. Find the directions and the factors by which it stretches in those directions.
4. Let B = {v1, v2, V3} and C = {w1, w2, w3} be bases for R3, with vectors defined below. -(:) -() --() v1 = V2 = V3 = and --(:) --E) --() 1 wi = , w2 = 1 , W3 = 1 Let L: R³ → R³ be the linear transformation defined by L: Find [L]B and [L]c, the matrices associated to L with respect to B and with respect to C.
Determine the matrix of the linear transformation T(x, y) = (3x + 5y, 3y – 2x) 5 -2 3 (a) -2 3 (b) (c) 3 3 3 (d) 3 3 -2 a 2]
5. Consider a linear transformation T: R2 → R2 which reflects a vector about the line y=-x, and dilates the reflected vector by a factor of 2. Find the standard matrix for the linear transformation.
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2. Let a = 2 ---- X= -2 4 and y = 4 2 (a) Find P, the projection matrix along the span of a. (b) Find proja (x). (c) Find proja (y). 3. Let A be a nonsingular 2 x 2 matrix. (a) Find P, the projection matrix onto C(A). (b) Find C(A) and comment on the result of part (a).
Let B = { 1 + x, 1 − x², 1+x+x²} and B' = {2 − x, 1 − x +x²,3 + 2x}. 3.) Find the transition matrix from B to B'. If [9] B' = -3 4 what is g as a polynomial?

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5. Let B= (b₁,b2), where b₁ = (1,3), b2 = (-3,1). (a) Specify the transition matrices Ps,B and PB,s (b) For T the linear transformation corresponding to a reflection in the line y = 3r, specify [TB.8.