10. Let T: R³ R4 and S R4 → R³ be defined by(a)(b)(c)T[x₁ + x₂\x2 + x30-3-0-and SX3 X1X3 X2X2X4x1 + x2 + x3 x4X1 X₂ X3 X4 --X1X2 X3 + X4Find the matrices of linear transformation for S and T.Find the matrix of linear transformation for ST, and give STIs ST invertible? Justify your answer.(E)

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Answered: 10. Let T: R³ R4 and S: R4 → R³ be…

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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The following is a linear transformation, find the matrix A such that T(X) = AX. T(X₁, X₂, x3) = (2x₁ + x3, 5x₁ + x₂)
IfT: P₁ → P₁ is a linear transformation such that T(1 + 5x) = 3 + 2x and T(2+ 9x) = -4 -2x, then T(4-2x)
(6) Let T : R² → R³ be the linear transformation defined by - (E)- [2(x1 – 02)] - 3x1 |x1 T X2 – x2 Xị + x2 (6a) Find the matrix A such that T(x) = Ax. %3D
2v1 5. (a) Let T be a linear transformation such that T What is its matrix? (b) Let T be a linear transformation such that T vi + 2v2. What is its matrix? %3D v3 + v1
8. Let T : R3 R³ be a transformation defined by T(x1,x2, x3) = (3x1 +5x3, x2, – x3). (a) Represent T as a matrix with T(x) = Ax. 1 (b) Determine whether b is in the range(T). 1
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2. Let T: R³ → R² be given by 3x1 + 2x2 4.x2 – 5x3 T I2 and let L: R² → R³ be given by (:)- 6x1 + 12 -3.12 (a) Show that T and L are linear transformations by showing that they are given by matrix multiplication. (b) Find the standard matrix for the composition Lo T.
Let T : P2 → P, be the linear transformation such that T(2x) = 2x? – 3x, T(0.5x – 4) = -2x – 4x + 3, T(4x² + 1) = –4x + 4. %3D Find T(1), T(x), T(x²), and T(ax² + bx + c), where a, b, and c are arbitrary real numbers. T(1) = T(x) = T(x²) = T(ax? + bx + c) =
Let the transformation T : IR³ →R² be given by the formula T(x,,X2,X3) = (2x, –x,, X, +3x;). Find the standard matrix of this transformation and write it in matrix multiplication form. Check correctness of your answer for the point (1,4,-3).
6. Determine if the linear transformation T(x1, x2, x3) = (2x1 — X2, —X1 − 2x2 + x3, x1 − 3x2 + x3) is (a) one-to-one (b) onto Hint: use your answer from problem 4.
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17. Find the 4 by 3 matrix A that represents a right shift: (x1,x2, x3) is transformed to (0, x1,x2, x3). Find also the left shift matrix B from R4 back to R³, transforming (x1,x2, x3, x4) to (x2, x3, x4). What are the products AB and BA?

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