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mxn matrix B. Show that if A is the standard matrix for T, then A= B. [Hint: Show that A and B have the same columns.] transformation Tonto

mxn matrix B. Show that if A is the standard matrix for T, then A= B. [Hint: Show that A and B have the same columns.] transformation Tonto

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
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80 CHAPTER 1 Linear Equations in Linear Algebra c. The standard matrix of a linear transformation from R² to R2 that reflects points through the horizontal axis, the vertical axis, or the origin has the form where a and d are ±1. d. A mapping T: R" → R" is one-to-one if each vector in R" maps onto a unique vector in Rm. e. If A is a 3 x 2 matrix, then the transformation X + Ax cannot map R2 onto R³. a [2] 0 d In Exercises 25-28, determine if the specified linear transforma- tion is (a) one-to-one and (b) onto. Justify each answer. 25. The transformation in Exercise 17 26/; 26 The transformation in Exercise 2 27. The transformation in Exercise 19 28. The transformation in Exercise 14 In Exercises 29 and 30, describe the possible echelon forms of the standard matrix for a linear transformation T. Use the notation of Example 1 in Section 1.2. 29. T: R³ R4 is one-to-one. →>>> 30. TR4 → R³ is onto. 31. Let T: R" → R" be a linear transformation, with A its standard matrix. Complete the following statement to make it true: "T is one-to-one if and only if A has pivot columns." Explain why the statement is true. [Hint: Look in the exercises for Section 1.7.] 32. Let T: R" → R" be a linear transformation, with A its standard matrix. Complete the following statement to make it true: "T maps R" onto Rm if and only if A has pivot columns." Find some theorems that explain why the statement is true. 33. Verify the uniqueness of A in Theorem 10. Let T: R" → Rm be a linear transformation such that T(x) = Bx for some 34. Why is the question "Is the linear transformation T onto?" an existence question? 35. If a linear transformation T: R"R" maps R" onto Rm can you give a relation between m and n? If T is one-to-one, what can you say about m and n? mxn matrix B. Show that if A is the standard matrix for T, then A= B. [Hint: Show that A and B have the same columns.] 36. Let S : RP→ R" and T: R"→ R" be linear transforma- tions. Show that the mapping x→ T(S(x)) is a linear trans- formation (from RP to Rm). [Hint: Compute T(S(cu + dv)) for u, v in RP and scalars c and d. Justify each step of the computation, and explain why this computation gives the desired conclusion.] [M] In Exercises 37-40, let T be the linear transformation whose standard matrix is given. In Exercises 37 and 38, decide if T is a one-to-one mapping. In Exercises 39 and 40, decide if T maps R5 onto R5. Justify your answers. 1024 tait 37. 39. 40. -5 10 -5 4 8 7 3-4 4-9 5 -3 5 4 -3-2 4 -7 3 7 5 6-8 5 12 -8 10 -8 -9 14 3 -5 4 2 -6 6 -6 -7 3 котор адфра -7 9 13 6 5 6 15 -7 -6 -8 -9 -5 -6 -89 14 -1 4 12 -5 -9 8 14 15 2 11 13 38. Г 7 5 10 6 12 8 12 -8 -6-2 9 ini ( 2mol adb davoni ziniog abal SA 191 4 -9 16 -4 7 5 Jonsd) ba tedi 1600 10
Transcribed Image Text:80 CHAPTER 1 Linear Equations in Linear Algebra c. The standard matrix of a linear transformation from R² to R2 that reflects points through the horizontal axis, the vertical axis, or the origin has the form where a and d are ±1. d. A mapping T: R" → R" is one-to-one if each vector in R" maps onto a unique vector in Rm. e. If A is a 3 x 2 matrix, then the transformation X + Ax cannot map R2 onto R³. a [2] 0 d In Exercises 25-28, determine if the specified linear transforma- tion is (a) one-to-one and (b) onto. Justify each answer. 25. The transformation in Exercise 17 26/; 26 The transformation in Exercise 2 27. The transformation in Exercise 19 28. The transformation in Exercise 14 In Exercises 29 and 30, describe the possible echelon forms of the standard matrix for a linear transformation T. Use the notation of Example 1 in Section 1.2. 29. T: R³ R4 is one-to-one. →>>> 30. TR4 → R³ is onto. 31. Let T: R" → R" be a linear transformation, with A its standard matrix. Complete the following statement to make it true: "T is one-to-one if and only if A has pivot columns." Explain why the statement is true. [Hint: Look in the exercises for Section 1.7.] 32. Let T: R" → R" be a linear transformation, with A its standard matrix. Complete the following statement to make it true: "T maps R" onto Rm if and only if A has pivot columns." Find some theorems that explain why the statement is true. 33. Verify the uniqueness of A in Theorem 10. Let T: R" → Rm be a linear transformation such that T(x) = Bx for some 34. Why is the question "Is the linear transformation T onto?" an existence question? 35. If a linear transformation T: R"R" maps R" onto Rm can you give a relation between m and n? If T is one-to-one, what can you say about m and n? mxn matrix B. Show that if A is the standard matrix for T, then A= B. [Hint: Show that A and B have the same columns.] 36. Let S : RP→ R" and T: R"→ R" be linear transforma- tions. Show that the mapping x→ T(S(x)) is a linear trans- formation (from RP to Rm). [Hint: Compute T(S(cu + dv)) for u, v in RP and scalars c and d. Justify each step of the computation, and explain why this computation gives the desired conclusion.] [M] In Exercises 37-40, let T be the linear transformation whose standard matrix is given. In Exercises 37 and 38, decide if T is a one-to-one mapping. In Exercises 39 and 40, decide if T maps R5 onto R5. Justify your answers. 1024 tait 37. 39. 40. -5 10 -5 4 8 7 3-4 4-9 5 -3 5 4 -3-2 4 -7 3 7 5 6-8 5 12 -8 10 -8 -9 14 3 -5 4 2 -6 6 -6 -7 3 котор адфра -7 9 13 6 5 6 15 -7 -6 -8 -9 -5 -6 -89 14 -1 4 12 -5 -9 8 14 15 2 11 13 38. Г 7 5 10 6 12 8 12 -8 -6-2 9 ini ( 2mol adb davoni ziniog abal SA 191 4 -9 16 -4 7 5 Jonsd) ba tedi 1600 10
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