Hi i need help with the problems highlighted and circled in red, its only linear algebra

176 CHAPTER 2 Matrices and Linear Transformations 4. Give the domain and codomain of the matrix transforma- In Exercises 25-34, linear transformations are given. Compute tion induced by AL. their standard matrices. Give the domain and codomain of the matrix transforma- tion induced by BY. 25 T: R2 - R2 defined by T ( ) = +x 6. Give the domain and codomain of the matrix transforma- 26. T: R2 - R2 defined by 7 ( ) =[ [2x1 + 3x2] tion induced by C 4X1 + 5x2 [ x 1 + x2 + x3] '7. Compute T 8. Compute TB 27. T : R3 - R2 defined by T = 2x1 3.x2 9. Compute To () 28. T: R2 - R3 defined by 7 (X ) = 2x1 - X2 10. Compute TA X1 + X2_ X1 - X2 29) T : R2 - R4 defined by 7 (4]) = 2x1 - 3X2 0 11 Compute TB 12. Compute Te *2 x1 - 2x3 30. T: R3 - R' defined by 1 = -3x1 + 4x2 13. Compute TA 14. Compute T 0 X1 - X2 0 ([-]) 31. T: R2 - R* defined by 7 (]) - 3x1 16. Compute T *2 [2x1 - x2 + 3x4] 32. T: R4 - R3 defined by 7 = -x1 + 2x4 17. Compute T's 18. Compute T ;]) 3X2 - X3 33 T: R3 - R3 defined by T(v) = v for all v in R3 34) T: R3 -> R' defined by T(v) = 0 for all v in R3 19. Compute TACT and TAF In Exercises 35-54, determine whether the state- TA ments are true or false. 35. Every function from R" to R" has a standard matrix. 36. Every matrix transformation is linear. 20. Compute A(e, ) and TA(e3). 37. A function from R" to R" that preserves scalar multipli- In Exercises 21-24, identify the values of n and m for each cation is linear. linear transformation T : R" - Rm. 38. The image of the zero vector under any linear transfor- mation is the zero vector. 21 T is defined by T = [x1 - x2] 39. If T: R3 - R2 is linear, then its standard matrix has size 3 x 2. 40. The zero transformation is linear. 22. T is defined by 7 (X - x1-x2 41. A function is uniquely determined by the images of the * 2 standard vectors in its domain. 42. The first column of the standard matrix of a linear trans- x1 - 4x2 formation is the image of the first standard vector under 23 T is defined by 7 () 2x1 - 3x2 the transformation. 0 X2 43. The domain of a function f is the set of all images f (x). 44. The codomain of any function is contained in its range. [5.x1 - 4x2 + X3 - 2x4] 45. If f is a function and f(u) = f(v), then u = v. 24. T is defined by 7 = -2x2 + 4x4 46. The matrix transformation induced by a matrix A is a 3.x1 - 5X3 linear transformation.2.4 The Inverse of a Matrix 143 If the reduced row echelon form of [A I,] is [R B], then ON B is an invertible matrix. 31 WN - - 63. If the reduced row echelon form of [A In] is [R B]. then AN- O-ON BA equals the reduced row echelon form of A. WHO - NHO $54 Suppose that A is an invertible matrix and u is a solution 32. - - N of AX = The solution of Ax = differs from u . O by 2p3, where p3 is the third column of A-! 33 WN 55. Prove directly that statement (a) in the Invertible Matrix Theorem implies statements (e) and (h). In Exercises 56-63, a system of linear equations is given. 34 INW - wood (a) Write each system as a matrix equation Ax = b. (b) Show that A is invertible, and find A-!. (c) Use A- to solve each system. T&F In Exercises 35-54, determine whether the state- 56. *1 + 2x2 = 9 -X1 - 3x2 = -6 ments are true or false. 2x1 + 3.x2 = 3 57. 2x1 + 5.12 = 4 A matrix is invertible if and only if its reduced row ech- lon form is an identity matrix. X1+ x2+ x3= 4 -x1+ 13 = -4 For any two matrices A and B, if AB = I, for some pos- 58. 2x1 + 12 + 4.13 = 7 59 X1 + 2x2 - 2x3 = 3 itive integer n, then A is invertible. 3x1 + 2x2 + 6.13 = -1 2x1 - *2 + 13 = 1 For any two n x n matrices A and B. if AB = I,, then X1+x2+13 =-5 2x1 + 3x2 -4x3 = -6 BA = In. 60. 2x1 + 12 + 13 = -3 61. -X1 - x2+ 2x3 = For any two n x n matrices A and B, if AB = I,, then A 3.x1 +x3 = 2 - 12 + 13 = 3 s invertible and A = B. If an n x n matrix has rank n, then it is invertible. - x3+x=3 If an n x n matrix is invertible, then it has rank n. 62. 2x1 - X2 - X3 -x1 + x2 + x3 + x4 = 4 A square matrix is invertible if and only if its reduced X2 + X3 + X4 =-1 row echelon form has no zero row. If A is an n x n matrix such that the only solution of X1 - 2x2 - X3+ 14 = 4 Ax = 0, then A is invertible. X1+ x2 - 14 = -2 43 63. An n x n matrix is invertible if and only if its columns -X1 - x2+ x3+*= are linearly independent. -3.x1 + *2 + 2x3 An n x n matrix is invertible if and only if its rows are inearly independent. 45 If a square matrix has a column consisting of all zeros, 64 Let A = 1 !] then it is not invertible. (a) Verify that A2 - 3A + 12 = 0. 6. If a square matrix has a row consisting of all zeros, then (b) Let B = 3/2 - A. Use B to prove that A is invertible it is not invertible. and B = A-1. Any invertible matrix can be written as a product of ele- nentary matrices. If A and B are invertible n x n matrices, then A + B is 65. Let A = 2 6 -?] invertible. (a) Verify that A3 - 542 + 9A - 413 = 0. 9 If A is an n x n matrix such that Ax = b is consistent for every b in R", then Ax = b has a unique solution for (b) Let B = -(A2 - 5A + 9/3). Use B to prove that A is every b in R". invertible and B = A-1. If A is an invertible n x n matrix and the reduced row (c) Explain how B in (b) can be obtained from the echelon form of [A B] is [, C], then C = B-A. equation in (a). 51 If the reduced row echelon form of [A 1,, ] is [R B], then 66. Let A be an n x n matrix such that A= = 1,. Prove that A B = A-. is invertible and A = A