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1. Let A = , and define T : R² → R² by T(x) = Ax. -6 10 %3D 2 -4 10. A = Find the images under T of u = 3 1 and v = 4. 10 8 3 2. Let A = a. u = and v = 11. Let b = and let A be the Define T : R in the range of the linear transforr why not? R' by T(x) = Ax. Find T(u) and T(v). %3D In Exercises 3-6, with T defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is unique. %3D 12. Let b = and let A be the 0 -3 6 ,b = 1 -1 3. A = -3 1 3 b in the range of the linear transfor why not? 2 -2 -1 1 -2 1 -3 ,b = 3 4. A = In Exercises 13-16, use a rectangular -5 6. u = V = and their images 1 -5 mation T. (Make a separate and reason exercise.) Describe geometrically wha in R?. 5. A = -3 %3D 7 1 -3 27 8 ,b 13. T(x) = 3 -8 6. A = %3D 1 3 1 8. 10 14. T(x) = 7. Let A be a 6 x 5 matrix. What must a and b be in order to define T: R -→ R° by T(x) = Ax? 15. T(x) = 8. How many rows and columns must a matrix A have in order to define a mapping from R$ into R’ by the rule T(x) = Ax? 16. T) =: :: 16. T(x): For Exercises 9 and 10, find all x in R that are mapped into the zero vector by the transformation x Ax for the given matrix A. 5 -57 17. Let T: R → R? be a linear tran [3 and maps v = into [1 -3 that T is linear to find the image 3101

1. Let A = , and define T : R² → R² by T(x) = Ax. -6 10 %3D 2 -4 10. A = Find the images under T of u = 3 1 and v = 4. 10 8 3 2. Let A = a. u = and v = 11. Let b = and let A be the Define T : R in the range of the linear transforr why not? R' by T(x) = Ax. Find T(u) and T(v). %3D In Exercises 3-6, with T defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is unique. %3D 12. Let b = and let A be the 0 -3 6 ,b = 1 -1 3. A = -3 1 3 b in the range of the linear transfor why not? 2 -2 -1 1 -2 1 -3 ,b = 3 4. A = In Exercises 13-16, use a rectangular -5 6. u = V = and their images 1 -5 mation T. (Make a separate and reason exercise.) Describe geometrically wha in R?. 5. A = -3 %3D 7 1 -3 27 8 ,b 13. T(x) = 3 -8 6. A = %3D 1 3 1 8. 10 14. T(x) = 7. Let A be a 6 x 5 matrix. What must a and b be in order to define T: R -→ R° by T(x) = Ax? 15. T(x) = 8. How many rows and columns must a matrix A have in order to define a mapping from R$ into R’ by the rule T(x) = Ax? 16. T) =: :: 16. T(x): For Exercises 9 and 10, find all x in R that are mapped into the zero vector by the transformation x Ax for the given matrix A. 5 -57 17. Let T: R → R? be a linear tran [3 and maps v = into [1 -3 that T is linear to find the image 3101

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Linear Transformations
Section6.1: Introduction To Linear Transformations
Problem 39E: For the linear transformation from Exercise 33, find a T(1,1), b the preimage of (1,1), and c the...
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1. Suppose T : R$ → R² and T (x) = Ax for some matrix A and for each x in R’. How many rows and columns does A have? 2. Let A = Give a geometric description of the transformation x + Ax. 3. The line segment from 0 to a vector u is the set of points of the form tu, where Ust<1. Show that a linear transformation T maps this segment into the segment between 0 and T (u). 1.8 EXERCISES 1. Let A = ,and define T : R² → R² by T (x) = Ax. 3 2 10 -6 -4 1 10. A = 2 Find the images under T of u = 3 1 and v = 1 4 10 8 3 -1 2. Let A = ,u = 6. and v = b 11. Let b = and let A be the matrix in Exercise 9. Is E Define T : R → R³ by T(x) = Ax. Find T(u) and T(v). in the range of the linear transformation x + Ax? Why on why not? In Exercises 3–6, with T defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is unique. 3 12. Let b = and let A be the matrix in Exercise 10. Is 1 0 -3 1 2 -2 -1 -1 3. А%3D -3 b = b in the range of the linear transformation x+ Ax? Why on why not? 1 -2 1 -3 3 -6 4. A = b = -4 In Exercises 13-16, use a rectangular coordinate system to plo 2 -5 6 -5 u = , and their images under the given transfor 5. A= =[ mation T. (Make a separate and reasonably large sketch for eacl exercise.) Describe geometrically what T does to each vector > in R2. 1 -3 1 -3 3 -8 8 ,b = 13. Т(х) — 6. A = 1 1 8 14. T(x) = 7. Let A be a 6 x 5 matrix. What must a and b be in order to define T: Rª → R³ by T(x) = Ax? 15. T(х) %3D 8. How many rows and columns must a matrix A have in order to define a mapping from R$ into R' by the rule T(x) = Ax? 16. T(х) 3 For Exercises 9 and 10, find all x in R* that are mapped into the zero vector by the transformation x Ax for the given matrix A. 17. Let T : R2 → R? be a linear transformation that maps u into and maps v = into Use the fa -3 -5 that T is linear to find the images under T of 2u, 3v, ar 1 -3 2u + 3v. 9. A = 2 -4 4 -4
Transcribed Image Text:1. Suppose T : R$ → R² and T (x) = Ax for some matrix A and for each x in R’. How many rows and columns does A have? 2. Let A = Give a geometric description of the transformation x + Ax. 3. The line segment from 0 to a vector u is the set of points of the form tu, where Ust<1. Show that a linear transformation T maps this segment into the segment between 0 and T (u). 1.8 EXERCISES 1. Let A = ,and define T : R² → R² by T (x) = Ax. 3 2 10 -6 -4 1 10. A = 2 Find the images under T of u = 3 1 and v = 1 4 10 8 3 -1 2. Let A = ,u = 6. and v = b 11. Let b = and let A be the matrix in Exercise 9. Is E Define T : R → R³ by T(x) = Ax. Find T(u) and T(v). in the range of the linear transformation x + Ax? Why on why not? In Exercises 3–6, with T defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is unique. 3 12. Let b = and let A be the matrix in Exercise 10. Is 1 0 -3 1 2 -2 -1 -1 3. А%3D -3 b = b in the range of the linear transformation x+ Ax? Why on why not? 1 -2 1 -3 3 -6 4. A = b = -4 In Exercises 13-16, use a rectangular coordinate system to plo 2 -5 6 -5 u = , and their images under the given transfor 5. A= =[ mation T. (Make a separate and reasonably large sketch for eacl exercise.) Describe geometrically what T does to each vector > in R2. 1 -3 1 -3 3 -8 8 ,b = 13. Т(х) — 6. A = 1 1 8 14. T(x) = 7. Let A be a 6 x 5 matrix. What must a and b be in order to define T: Rª → R³ by T(x) = Ax? 15. T(х) %3D 8. How many rows and columns must a matrix A have in order to define a mapping from R$ into R' by the rule T(x) = Ax? 16. T(х) 3 For Exercises 9 and 10, find all x in R* that are mapped into the zero vector by the transformation x Ax for the given matrix A. 17. Let T : R2 → R? be a linear transformation that maps u into and maps v = into Use the fa -3 -5 that T is linear to find the images under T of 2u, 3v, ar 1 -3 2u + 3v. 9. A = 2 -4 4 -4
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