2. Suppose that T: R² R² is a linear transformation and that T(8) - ]=Find the standard matrix for T, that is, the matrix A such thatT(-4)=T() Au for any 7 € R².=and

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Answered: ¹ (8) -] = Find the standard matrix for…

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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-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 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.
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9. Consider the following 2 x 2 matrix A and basis S of R²: A ¹ = [² %) and 5 = {[_¹₂] · [_³;]} The matrix A defines a linear operator on R². Find a matrix B that represents the mapping A relative to the basis S.
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Could you explain how to show this in detail?
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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¹ (8) -] = Find the standard matrix for T, that is, the matrix A such that Suppose that T: R² R² is a linear transformation and that T ->> T T(7) Au for any 7 € R². = and