Linear Transformations, Matrix Algebra

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<ul><li> 1. AnnouncementsQuiz 2 on Wednesday Jan 27 on sections 1.4, 1.5, 1.7 and 1.8If you have any grading issues with quiz 1, please discuss withme asap.Solution to quiz 1 will be posted on the website by Monday. </li></ul> <p> 2. Last Class... A transformation (or function or mapping) T from Rn to Rm is arule that assigns to each vector x in Rn a vector T (x) in Rm . 3. Last Class... A transformation (or function or mapping) T from Rn to Rm is arule that assigns to each vector x in Rn a vector T (x) in Rm . The set Rn is called Domain of T . 4. Last Class... A transformation (or function or mapping) T from Rn to Rm is arule that assigns to each vector x in Rn a vector T (x) in Rm . The set Rn is called Domain of T . The set Rm is called Co-Domain of T . 5. Last Class... A transformation (or function or mapping) T from Rn to Rm is arule that assigns to each vector x in Rn a vector T (x) in Rm . The set Rn is called Domain of T . The set Rm is called Co-Domain of T . The notation T : Rn Rm means the domain is Rn and the co-domain is Rm . 6. Last Class... A transformation (or function or mapping) T from Rn to Rm is arule that assigns to each vector x in Rn a vector T (x) in Rm . The set Rn is called Domain of T . The set Rm is called Co-Domain of T . The notation T : Rn Rm means the domain is Rn and the co-domain is Rm . For x in Rn , the vector T (x) is called the image of x. 7. Last Class... A transformation (or function or mapping) T from Rn to Rm is arule that assigns to each vector x in Rn a vector T (x) in Rm . The set Rn is called Domain of T . The set Rm is called Co-Domain of T . The notation T : Rn Rm means the domain is Rn and the co-domain is Rm . For x in Rn , the vector T (x) is called the image of x. Set of all images T (x) is called the Range of T . 8. Linear TransformationA transformation (or function or mapping) is Linear if 9. Linear TransformationA transformation (or function or mapping) is Linear ifT (u + v) = T (u) + T (v) for all u and v in the domain of T . 10. Linear TransformationA transformation (or function or mapping) is Linear ifT (u + v) = T (u) + T (v) for all u and v in the domain of T .T (c u) = cT (u) for all u and all scalars c . 11. ImportantIf T is a linear transformation 12. ImportantIf T is a linear transformation T (0) = (0). 13. ImportantIf T is a linear transformation T (0) = (0). T (c u + d v) = cT (u) + dT (v) for all u and v in the domain of T. 14. Interesting Linear Transformations0 1 3 1Let A =1 0 u=2,v = 3Let T : R2 R2 a linear transformation dened by T (x) = Ax. Findthe images under T of u, v and u+v.Solution: Image under T of u and v is nothing butT (u) = 0 1 3 = 0.1.+ +12.2 = 21 02 3 ( ) 3 0.30 1 1 0.1 + (1).3 3T (v) = 1 0 3 = 1.1 + 0.3 = 1 15. Interesting Linear Transformations 3 1 4Since u+v = + = ,2 3 5 The image under T ofu+v is nothing but T (u+v) = 0 11 045=0.4 + (1).51. 4 + 0. 5= 5 4The next picture shows what happened here. 16. Rotation TransformationHere T rotates u, v and u+vcounterclockwise about the origin through 900 . yx0 17. Rotation TransformationHere T rotates u, v and u+vcounterclockwise about the origin through 900 . yux0 18. Rotation TransformationHere T rotates u, v and u+vcounterclockwise about the origin through 900 . yT (u) ux0 19. Rotation TransformationHere T rotates u, v and u+vcounterclockwise about the origin through 900 . yT (u)vux0 20. Rotation TransformationHere T rotates u, v and u+vcounterclockwise about the origin through 900 . yT (u)vuT (v) x0 21. Rotation TransformationHere T rotates u, v and u+vcounterclockwise about the origin through 900 . yu+v T (u)vuT (v) x0 22. Rotation TransformationHere T rotates u, v and u+vcounterclockwise about the origin through 900 . yu+v T (u+v)T (u)vuT (v) x0 23. Rotation TransformationHere T rotates u, v and u+vcounterclockwise about the origin through 900 . yu+v T (u+v)T (u)vuT (v) x0 24. Rotation TransformationHere T rotates u, v and u+vcounterclockwise about the origin through 900 . yu+vT T (u+v)T (u)vuT (v) x0 25. Interesting Linear Transformations0 131Let A =1 0 u=2,v =3Let T : R2 R2 a linear transformation dened by T (x) = Ax. Findthe images under T of u and vSolution: Image under T of u and v is nothing butT (u) = 0 1 3 = 0133 + 1)22 = 21 0 2 . +( ..0.30 1 1 0.1 + (1).3 3T (v) = 1 0 3 = 1.1 + 0.3 = 1 26. Reection TransformationHere T reects u and v about the line x = y . yx0 27. Reection TransformationHere T reects u and v about the line x = y . yux0 28. Reection TransformationHere T reects u and v about the line x = y . yTuux0 29. Reection TransformationHere T reects u and v about the line x = y . yTuux0 30. Reection TransformationHere T reects u and v about the line x = y . yv Tuux0 31. Reection TransformationHere T reects u and v about the line x = y . yv TuuTvx0 32. Reection TransformationHere T reects u and v about the line x = y . yv TuuTvx0 33. Reection TransformationHere T reects u and v about the line x = y . yv TuuTvx0 34. Example 6, Section 1.8 1 2 11 3 4 5 9Let A = , b = 0 1 1 3 3 54 6Let T be dened by by T (x) = Ax. Find a vector x whose imageunder T is b and determine whether x is unique. 35. Example 6, Section 1.8 1 2 11 3 4 5 9Let A = , b = 0 1 1 3 3 54 6Let T be dened by by T (x) = Ax. Find a vector x whose imageunder T is b and determine whether x is unique.Solution The problem is asking you to solve Ax = b. In other words,write the augmented matrix and solve. 36. 12 11 R2-3R1 34 59 R4+3R1 0113 3 54 6 12 1 1 02 26 = 01 13 01 13 37. Divide row 2 by 2 1 2 1 1 0 1 13 = 0 1 13 0 1 13 1 211 0 1 13 R3-R2 R4+R2 0 1 13 0 1 1 3 38. 1 2 1 1 0 11 3 0 00 0 0 00 0 39. 12 1 1 0 1 1 3 0 0 0 00 0 0 0 Since column 3 doesnot have a pivot, x3 is a free variable. We can solve for x1 and x2 in terms of x3 . x1 2x2 + x3 = 1x2+ x3 = 3 We have x2 = 3 x3 and x1 = 1 + 2x2 x3 = 1 + 2(3 x3 ) x3 = 7 3x3 . 40. The solution is thusx1 7 3x3 x= x2 =3 x3 x3 x3 Since we can choose any value for x3 , the solution is NOT unique. 41. Example 10, Section 1.813 92 10 3 4Let A = Find all x in R4 that are mapped into 0 1 23 2 3 05the zero vector by the transformation x Ax for the given matrixA. 42. Example 10, Section 1.813 92 10 3 4Let A = Find all x in R4 that are mapped into 0 1 23 2 3 05the zero vector by the transformation x Ax for the given matrixA.Solution The problem is asking you to solve Ax = 0. In other words,write the augmented matrix for the homogeneous system and solve. 43. 1 3 9 2 0R2-R1 1 0 3 40R4+2R1 0 1 2 3 0 23 0 5 0 1 3 92 0 0 3 6 6 0 = 0 1 23 0 0 9 1890 44. Divide row 2 by -3 and row 4 by 91 3 9 2 0 0 1 2 2 0 = 0 1 2 3 0 0 1 2 1 0 1 3 9 20 0 1 2 2 0 R3-R2R4-R2 0 1 2 30 0 1 2 10 45. 1 3 92 0 0 1 22 0 0 0 01 0 R4+R3 0 0 010 1 3 9 2 0 0 1 2 2 0 = 0 0 0 1 0 0 0 0 0 0 46. How many pivot columns? 47. How many pivot columns? 3. Columns 1,2 and 4. Which is the free variable? 48. How many pivot columns? 3. Columns 1,2 and 4. Which is the free variable? x3 . Write the system of equations so that we can express the basic variables in terms of the free variables. x1 + 3x2 + 9x3 + 2x4 = 0 x2 + 2x3 + 2x4 = 0x4 = 0 Thus, x2 = 2x3 and x1 = 3x2 9x3 = 3(2x3 ) 9x3 = 3x3 . Our solution is thusx1 3x33 x= x2 2x = 2 = x3 3 x3 x3 1 x4 0 0 49. Chapter 2 Matrix Algebra DenitionDiagonal Matrix: A square matrix (same number of rows andcolumns) with all non-diagonal entries 0.Example1 0 0 0 9 0 0 0 7 0 0 , 0 0 0 0 0 4 0 0 0 1 0 0 0 3 50. Chapter 2 Matrix Algebra DenitionZero Matrix: A matrix of any size with all entries 0.Example 0 0 0 0 0 0 0 0 0 0 0 0 , 0 00 0 0 0 0 0 0 0 0 0 0 0 51. Matrix Addition Two matrices are equal ifthey have the same sizethe corresponding entries are all equal 52. Matrix Addition Two matrices are equal if they have the same size the corresponding entries are all equalIf A and B are m n matrices, the sum A + B is also an m n matrix 53. Matrix Addition Two matrices are equal if they have the same size the corresponding entries are all equalIf A and B are m n matrices, the sum A + B is also an m n matrix The columns of A + B is the sum of the corresponding columns of Aand B . 54. Matrix Addition Two matrices are equal if they have the same size the corresponding entries are all equalIf A and B are m n matrices, the sum A + B is also an m n matrix The columns of A + B is the sum of the corresponding columns of Aand B .A + B is dened only if A and B are of the same size. 55. Matrix AdditionLet1 2 30 1 3 0 1 A= 2 3 4 ,B = 2 0 4 ,C = 2 0 3 4 50 0 5 0 0Find A + B , A + C and B + CSolution 1+0 2+1 3+3 1 3 6 A+B = 2+2 3+0 4+4=4 3 8 3+0 4+0 5+5 3 4 10 56. Matrix AdditionLet1 2 30 1 3 0 1 A= 2 3 4 ,B = 2 0 4 ,C = 2 0 3 4 50 0 5 0 0Find A + B , A + C and B + CSolution 1+0 2+1 3+3 1 3 6 A+B = 2+2 3+0 4+4=4 3 8 3+0 4+0 5+5 3 4 10Both A + C and B + C are not dened since they are of dierentsizes. 57. Scalar MultiplicationIf r is a scalar (number) then the scalar multiple rA is the matrixwhose columns are r times the columns in A. Let1 2 30 1 A= 2 3 4,C = 2 0 3 4 50 0Find 4A and 2CSolution4 8 12 4A = 8 12 16 12 16 20 02 2C = 4 00 0 58. Basic Algebraic Properties For all matrices A, B and C of the same size and all scalars r and sA+B = B +A(A + B ) + C = A + (B + C )A+0 = Ar (A + B ) = rA + rB(r + s )A = rA + sAr (sA) = (rs )A </p>