Find the matrix M of the linear transformation T: R³ → R² given by(E)-[X2M =T-9x1+x₂+(-4)x3-7x1+3x₂[_-4)²³].9-

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Answered: Find the matrix M of 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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6. Show that T is a linear transformation. T(x1, x2) = (2x2 – 3x1, x1 4x2, 0, x₂).
Let z be a real number and let A be the matrix 2 I A -- ( 34 ). -1 -2 i. Find all values of r such that A is invertible. ii. Find all values of r such that A is its own inverse.
Let a and B be the standard bases for R" and R" respectively. For each of the following linear transformations T; : R² → R", compute the matrix [T;%- 2a b () (a) T1 : R² → R³, T За + 46 a () - (*:) (C) (C) (2а + 3b —. a +b (b) Т, : R3 — R?, T, b (c) T3 : R³ → R', T3 = (2a + b – 3c). (d) T4 : R" → R", T4
Each of J, K, L, M and N is a linear transformation from R2 to R². These functions are given as follows: Jx1, х2) %3D (Зхі — 5х2, —6х1 + 10x), K(x1, x2) = (-V3x2, 3x1), L(x1, x2) = (x2, –x1), M(x1, x2) = (Зх1 + 5х, бх1 — бх2), N(x1, x2) = (-v5x1, v5x2). (a) In each case, compute the determinant of the transformation. . det J= ,det K ,det L= ,det M= ,det N
Find foo fo x² (x²+4) (x²+2x+2) x² +24+2) d dx
Let T: R² R² be defined by T -{[5].[3]} C = Given Pc = [ [T](PB (u)) 1 2 = 3x1x2 -3 ([2₂])-[¹22] Lu= [2] B-{].G]}, and [3]} = Let -x1 + Ex: 5 x1 = -2 , use the Fundamental Theorem of Matrix Representations to find [T](PB(u)). 5
Find the standard matrix for the linear transformation T(1, 9) — (2а — Зу, т — у, у — 4г) -3 1 -1 1 1 -4 2 1 -4 -3 -1 1 -3 -1 2 1 1 -3 -1 2.
Let T1 and T2 be linear transformations given by X1 ЗХ1 + 5x2 X2 -2x1 + 8x2 X1 -2x1 + 9x2 X2 5x2 Find the matrix A such that the following is true. (a) T1(T2(x)) = Ax A = (b) T2(T1(x)) = Ax A = (c) T1(T1(x)) = Ax A = (d) T2(T2(x)) = = Ax A =
b) Evaluate det[(2A)-1 (3A")], where A is any invertible (2 x 2) matrix.
[:] Suppose that T is a linear transformation from ³ into ² such that 3 -[-2]. (₂) - [8].a T -[2], and T(3) = [1] The columns of k T(e₁) O *1 Tx2 = 2x1 x3 *1 TX2 002 1 0 0 0 Tx2 = *3 3x1-2x2] Tx2 = 2x1 001 Find a formula for the image of an arbitrary x= x2 in ³. 4x2 + x3 1 0 are e₁ 3x1 + 2x2 - 4x3 -2x1 + x3 3x1-2x1 2x1 3x1 + 2x2-4x3 -2x1 wwww + X3 x3 0 e2 ез 0 akan wwwwww.
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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.

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Find the matrix M of the linear transformation T : R3 → R2 given by () [ = M: T -9x1 + x2 + (-4)x3 -721 +3x2 (51)23] 9. 지