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Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Dugh-л/4 √2).] at maps ei that leaves =/4 radian horizontal izontal x₁- XI. that trans- nd then re- Cal x₂-axis cts points gh the X2- r transfor- = the angle ely a rota- ion? that T (e₁) Using the 15. 16. ? ? ? ? ? ? ? ? I ? ? X1 ? H X2 ? X3 X1 X2 = = 3x1 4x1 x₁ - x2 + x3 đi – 12 -2x1 + x₂ XI In Exercises 17-20, show that T is a linear transformation by finding a matrix that implements the mapping. Note that x₁, x₂,... are not vectors but are entries in vectors. 17. T(X1, X2, X3, X4) = (0, x₁ + x2, x2 + x3, x3 + x^) 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: R → 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 , then T is a linear transformation. c. When two linear transformations are performed one after another, the combined effect may not always be a linear transformation. d. A mapping T: R" → R" is onto R" if every vector x in R" maps onto some vector in Rm. e. If A is a 3 x 2 matrix, then the transformation x→ Ax cannot be one-to-one.
Transcribed Image Text:Dugh-л/4 √2).] at maps ei that leaves =/4 radian horizontal izontal x₁- XI. that trans- nd then re- Cal x₂-axis cts points gh the X2- r transfor- = the angle ely a rota- ion? that T (e₁) Using the 15. 16. ? ? ? ? ? ? ? ? I ? ? X1 ? H X2 ? X3 X1 X2 = = 3x1 4x1 x₁ - x2 + x3 đi – 12 -2x1 + x₂ XI In Exercises 17-20, show that T is a linear transformation by finding a matrix that implements the mapping. Note that x₁, x₂,... are not vectors but are entries in vectors. 17. T(X1, X2, X3, X4) = (0, x₁ + x2, x2 + x3, x3 + x^) 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: R → 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 , then T is a linear transformation. c. When two linear transformations are performed one after another, the combined effect may not always be a linear transformation. d. A mapping T: R" → R" is onto R" if every vector x in R" maps onto some vector in Rm. e. If A is a 3 x 2 matrix, then the transformation x→ Ax cannot be one-to-one.
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