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17. T(X₁, X2, X3, X4) = (0, X₁ + X2, X2 + X3, X3 + X4)

17. T(X₁, X2, X3, X4) = (0, X₁ + X2, X2 + X3, X3 + X4)

Advanced Engineering Mathematics
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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In Exercises 25-28, determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. 25. The transformation in Exercise 17
2 4 es n 1 In Exercises 15 and 16, fill in the missing entries of the matrix, assuming that the equation holds for all values of the variables. 15. 16. ? ? ? ? ? 22 ? ? ? ? ? ? ? XI D-L X3 X1 X2 = 3x₁ - 2x3 4x1 x1 - x2 + x3 X1 X2 -2x1 + x₂ X1 In Exercises 17-20, show that T is a linear transformation by finding a matrix that implements the mapping. Note that x₁, x.. are not vectors but are entries in vectors. 17. T(X1, X2, X3, X4) = (0, X₁ + X₂, X₂ + x3, x3 + x4) 18. T(x1, x₂) = (2x₂-3x₁, x₁ - 4x2, 0, X₂) 19. T(X₁, X2, X3) = (x₁ - 5x₂ + 4x3, x2 - 6x3) 20. T(X1, X2, X3, X4) = 2x₁ + 3x3 - 4x4 (T: R4 → R) 21. Let T: R² R² be a linear transformation such that T(x₁, x₂) = (x₁ + x₂, 4x₁ + 5x2). Find x such that T(x) (3,8). = 22. Let T: R² R³ be a linear transformation such that T(x₁.x₂) = (x₁ - 2x2,-x1 + 3x2, 3x1 - 2x2). Find x such that T(x) = (-1,4,9). In Exercises 23 and 24, mark each statement True or False. Justify each answer. 23. a. A linear transformation T: R" → R" is completely de- termined by its effect on the columns of the n x n identity matrix. b. If T: R² R² rotates vectors about the origin through an angle o, then T is a linear transformation,
Transcribed Image Text:2 4 es n 1 In Exercises 15 and 16, fill in the missing entries of the matrix, assuming that the equation holds for all values of the variables. 15. 16. ? ? ? ? ? 22 ? ? ? ? ? ? ? XI D-L X3 X1 X2 = 3x₁ - 2x3 4x1 x1 - x2 + x3 X1 X2 -2x1 + x₂ X1 In Exercises 17-20, show that T is a linear transformation by finding a matrix that implements the mapping. Note that x₁, x.. are not vectors but are entries in vectors. 17. T(X1, X2, X3, X4) = (0, X₁ + X₂, X₂ + x3, x3 + x4) 18. T(x1, x₂) = (2x₂-3x₁, x₁ - 4x2, 0, X₂) 19. T(X₁, X2, X3) = (x₁ - 5x₂ + 4x3, x2 - 6x3) 20. T(X1, X2, X3, X4) = 2x₁ + 3x3 - 4x4 (T: R4 → R) 21. Let T: R² R² be a linear transformation such that T(x₁, x₂) = (x₁ + x₂, 4x₁ + 5x2). Find x such that T(x) (3,8). = 22. Let T: R² R³ be a linear transformation such that T(x₁.x₂) = (x₁ - 2x2,-x1 + 3x2, 3x1 - 2x2). Find x such that T(x) = (-1,4,9). In Exercises 23 and 24, mark each statement True or False. Justify each answer. 23. a. A linear transformation T: R" → R" is completely de- termined by its effect on the columns of the n x n identity matrix. b. If T: R² R² rotates vectors about the origin through an angle o, then T is a linear transformation,
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