7. Given T:P, → P2, where T(a, + a,x) = a1 + (a, + a,)x + (a, + 2ao)x² is a lineartransformation & let B = {1+x,1 – 2x}amd C = {1,x, x²}be bases for B & C,respectivelya) Find the matrix of T with respect to bases B & Cb) Find [p(x)]e if p(x) = 10 +xnote: You must use augmented matrices, when necessaryc) Use matrix multiplication to find [T(p(x))]cd) Use [T(p(x))]c to find T(p(x))

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Answered: 7. Given T:P, → P2, where T(a, + a,x) =…

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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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.
-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 =
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
O Find the transformation matrix that wotates the axis X of a rec tangu lar coordinale system 60 toward X2 around the X3-axis
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.
7. Is there a linear transformation T : R³ → R³ such that T If so, what is its matrix?
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, ->
Please explain
Let L: R3 R2 be defined by the following. X1 + X3 ***] 8x2 Suppose X1 L X₂ P = = 1 -(HH) 4 3 3 1 2 B₁ = 4 ={[8][;}} is an ordered basis for the domain and B₂ = is an ordered basis for the range. Find the matrix representation P for L relative to B₁ and B₂ such that [L(u)]b₂ = P[u]ß₁'
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7. Given T:P, → P2, where T(a, + a,x) = a1 + (a, + a,)x + (a, + 2ao)x² is a linear transformation & let B = {1+x,1 – 2x}amd C = {1,x,x²}be bases for B & C,respectively a) Find the matrix of T with respect to bases B & C