The matrices4₁ = [18] · 4₂ = [od] ·,A3 =[0].A0-[9]0A4 =form a basis for the linear space V = R2x2. Write the matrix of the linear transformation T: R2x2relative to this basis.2x2→ R2such that T(A) = 4A +15AT

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Answered: The matrices 4₁ = [18] · A₂ = [od] · ,…

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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Find the matrix of the linear transformation T from R? R?, where T([1, 0]) = [1, -2], and T([2, 1]) = [2, 3] (Note: matrix A = [T([1,0]), T([0, 1])] ). 1 (a) 1 (b) 1. (c) -2 1 -2 3 -2 1 (d) 0. 3
Find the change of basis matrix [B]• 2 -{[0]·[15]} = { [0]·[4]} C 4 B
The standard basis & (e₁,e₂) and two custom bases B (b₁,b₂} and C (c₁, c₂} for R² are shown in the figures below. Standard basis S= (₁, ₂) 4 a. Find the change of basis matrix from custom B-coordinates to standard S-coordinates. [id] = id= -2 b Find the change of basis matrix from custom C-coordinates to standard S-coordinates [id] = -1 -1 -1 c. Find the change of basis matrix from B-coordinates to C-coordinates -3 -1/12 1/4 d. Find the change of basis matrix from C-coordinates to B-coordinates. 3 2 1 -1 -2 -9 b2 -3-2-1 82 el bl 1 2 3 b1 X Custom basis B = {b₁,b₂) |id|s [id Standard basis S= (0₁,0₂} a 3 2 1 -1 -2 -9 c2 -3 -2 -1 G2 62 217 el [id] ↑ cl 12 2 3 61 H x Custom basis C (C₁, C₂)
3. Find a matrix P that diagonalizes the matrix -3 9 18-3 4. Let B be the a basis of R², and let L: R² R² be the linear operator with the matrix [-3-6]
Let B = {b₁,b2,b3} be a basis for vector space V. Let T: V → V be a linear transformation with the following properties. == T(b₁) = -6b₁ + 5b₂, T(b₂) -3b₁ +4b₂, T (b3) =b₁ 1 Find [T]B, the matrix for T relative to B.
Show that T(x, y, z) = (4x + 2y – 2z, −2x + y + 3z, x - y - 2z) is not a one-to-one transformation from R³ to R³. Find a basis of the kernel of this transformation.
Let B= (b,, b,, b,} and D= (d,, d,} be bases for vector spaces V and W, respectively. Let T :V- W be a linear transformation with the following properties. T(b,) = - 3d, +2d2, T(b2) = - 4d, + 4d2, T(b3) = 5d2 Find the matrix for T relative to B and D. The matrix for T relative to B and D isO.
Explain why S is not a basis for P,. S = {1 – x, 1 – x2, -1 – 7x + 8x²} O sis linearly dependent. O s does not span P,. O S is linearly dependent and does not span P,.
Given the following system of linear equations: Зх, + 4х, 1 2.x, 2х, + X3 -1. 2x, 2х, + 4х, (a) Describe v, A and b if the convention used is y. A = b. (b) Determine the linear transformation T that A represents by the convention (v)T = v · A. (c) Find the standard matrix A' of Tif the convention T(v) = A' -v is used instead.
Find the stable vector of this Matrix: 0 0 1 0 1/2 1/2 1/3 2/3 0

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The matrices 4₁ = [18] · A₂ = [od] · , A3 = [0].A 0 -[9] 0 A4 = → R2 form a basis for the linear space V = R2x2. Write the matrix of the linear transformation T: R2X2 2x2 such that T(A) = 4A +15AT relative to this basis.