linear algebra - chapter 1 [yr2005]

1. Linear Algebra - Chapter 1 [YR2005] 1

3. Linear Algebra - Chapter 1 [YR2005] 3 Introduction ~ Matrices Information in science and mathematics is often organized into rows and columns to form rectangular arrays. Tables of numerical data that arise from physical observations Example: (to solve linear equations) Solution is obtained by performing appropriate operations on this matrix

4. 1.1 Introduction to Systems of Linear Equations

5. Linear Algebra - Chapter 1 [YR2005] 5 Linear Equations In x y variables (straight line in the xy-plane) where a1, a2, & b are real constants, In n variables where a1, …, an & b are real constants x1, …, xn = unknowns. Example 1 Linear Equations The equations are linear (does not involve any products or roots of variables).

6. Linear Algebra - Chapter 1 [YR2005] 6 Linear Equations The equations are not linear. A solution of is a sequence of n numbers s1, s2, ..., sn ? they satisfy the equation when x1=s1, x2=s2, ..., xn=sn (solution set). Example 2 Finding a Solution Set 1 equation and 2 unknown, set one var as the parameter (assign any value) or 1 equation and 3 unknown, set 2 vars as parameter

7. Linear Algebra - Chapter 1 [YR2005] 7 Linear Systems / System of Linear Equations

8. Linear Algebra - Chapter 1 [YR2005] 8 Linear Systems

9. Linear Algebra - Chapter 1 [YR2005] 9 Linear Systems In general: Every system of linear equations has either no solutions, exactly one solution, or infinitely many solutions. An arbitrary system of m linear equations in n unknowns: a11x1 + a12x2 + ... + a1nxn = b1 a21x1 + a22x2 + ... + a2nxn = b2 am1x1 + am2x2 + ... + amnxn = bm x1, ..., xn = unknowns, a’s and b’s are constants aij, i indicates the equation in which the coefficient occurs and j indicates which unknown it multiplies

10. Linear Algebra - Chapter 1 [YR2005] 10 Augmented Matrices Example: Remark: when constructing, the unknowns must be written in the same order in each equation and the constants must be on the right.

11. Linear Algebra - Chapter 1 [YR2005] 11 Augmented Matrices Basic method of solving system linear equations Step 1: multiply an equation through by a nonzero constant. Step 2: interchange two equations. Step 3: add a multiple of one equation to another. On the augmented matrix (elementary row operations): Step 1: multiply a row through by a nonzero constant. Step 2: interchange two rows. Step 3: add a multiple of one equation to another.

12. Linear Algebra - Chapter 1 [YR2005] 12 Elementary Row Operations (Example) r2= -2r1 + r2 r3 = -3r1 + r3

13. Linear Algebra - Chapter 1 [YR2005] 13 Elementary Row Operations (Example) r2 = ½ r2 r3 = -3r2 + r3 r3 = -2r3

14. Linear Algebra - Chapter 1 [YR2005] 14 Elementary Row Operations (Example) r1 = r1 – r2 r1 = -11/2 r3 + r1 r2 = 7/2 r3 + r2 Solution:

15. 1.2 Gaussian Elimination

16. Linear Algebra - Chapter 1 [YR2005] 16 Echelon Forms Reduced row-echelon form, a matrix must have the following properties: If a row does not consist entirely of zeros the the first nonzero number in the row is a 1 = leading 1 If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix. In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row. Each column that contains a leading 1 has zeros everywhere else.

17. Linear Algebra - Chapter 1 [YR2005] 17 Echelon Forms A matrix that has the first three properties is said to be in row-echelon form. Example: Reduced row-echelon form: Row-echelon form:

18. Linear Algebra - Chapter 1 [YR2005] 18 Elimination Methods Step 1: Locate the leftmost non zero column Step 2: Interchange r2 ? r1. Step 3: r1 = ½ r1. Step 4: r3 = r3 – 2r1.

19. Linear Algebra - Chapter 1 [YR2005] 19 Elimination Methods Step 5 : continue do all steps above until the entire matrix is in row-echelon form. r2 = -½ r2 r3 = r3 – 5r2 r3 = 2r3

20. Linear Algebra - Chapter 1 [YR2005] 20 Elimination Methods Step 6 : add suitable multiplies of each row to the rows above to introduce zeros above the leading 1’s. r2 = 7/2 r3 + r2 r1 = -6r3 + r1 r1 = 5r2 + r1

21. Linear Algebra - Chapter 1 [YR2005] 21 Elimination Methods 1-5 steps produce a row-echelon form (Gaussian Elimination). Step 6 is producing a reduced row-echelon (Gauss-Jordan Elimination). Remark: Every matrix has a unique reduced row-echelon form, no matter how the row operations are varied. Row-echelon form of matrix is not unique: different sequences of row operations can produce different row- echelon forms.

22. Linear Algebra - Chapter 1 [YR2005] 22 Back-substitution Bring the augmented matrix into row-echelon form only and then solve the corresponding system of equations by back-substitution. Example: [Solved by back substitution]

23. Linear Algebra - Chapter 1 [YR2005] 23 Back-Substitution Step 3. Assign arbitrary values to the free variables [parameters], if any

24. Linear Algebra - Chapter 1 [YR2005] 24 Homogeneous Linear Systems A system of linear equations is said to be homogeneous if the constant terms are all zero. Every homogeneous sytem of linear equations is consistent, since all such systems have x1=0,x2=0,...,xn=0 as a solution [trivial solution]. Other solutions are called nontrivial solutions.

25. Linear Algebra - Chapter 1 [YR2005] 25 Homogeneous Linear Systems Example: [Gauss-Jordan Elimination]

26. Linear Algebra - Chapter 1 [YR2005] 26 Homogeneous Linear Systems The corresponding system of equations is Solving for the leading variables yields The general solution is The trivial solution is obtained when s=t=0

27. Linear Algebra - Chapter 1 [YR2005] 27 Homogeneous Linear Systems Theorem: A homogeneous system of linear equations with more unknowns than equations has infinitely many solutions.

28. 1.3 Matrices and Matrix Operations

29. Linear Algebra - Chapter 1 [YR2005] 29 Matrix Notation and Terminology A matrix is a rectangular array of numbers with rows and columns. The numbers in the array are called the entries in the matrix. Examples: The size of a matrix is described in terms of the number of rows and columns its contains. A matrix with only one column is called a column matrix or a column vector. A matrix with only one row is called a row matrix or a row vector.

30. Linear Algebra - Chapter 1 [YR2005] 30 Matrix Notation and Terminology aij = (A)ij = the entry in row i and column j of a matrix A. 1 x n row matrix a = [a1 a2 ... an] m x 1 column matrix A matrix A with n rows and n columns is called a square matrix of order n. Main diagonal of A = {a11, a22, ..., ann}

31. Linear Algebra - Chapter 1 [YR2005] 31 Operations on Matrices Definition: Two matrices are defined to be equal if they have the same size and their corresponding entries are equal. If A = [aij] and B = [bij] have the same size, then A=B if and only if (A)ij=(B)ij, or equivalently aij=bij for all i and j. Definition: If A and B are matrices of the same size, then the sum A+B is the matrix obtained by adding the entries of B to the corresponding entries of A, and the difference A–B is the matrix obtained by subtracting the entries of B from the corresponding entries of A. Matrices of different sizes cannot be added or subtracted.

32. Linear Algebra - Chapter 1 [YR2005] 32 Operations on Matrices If A = [aij] and B = [bij] have the same size, then (A+B)ij = (A)ij + (B)ij = aij + bij and (A-B)ij = (A)ij – (B)ij = aij - bij Definition: If A is any matrix and c is any scalar, then the product cA is the matrix obtained by multiplying each entry of the matrix A by c. The matrix cA is said to be a scalar multiple of A. If A = [aij], then (cA)ij = c(A)ij = caij.

33. Linear Algebra - Chapter 1 [YR2005] 33 Operations on Matrices Definition If A is an mxr matrix and B is an rxn matrix, then the product AB is the mxn matrix whose entries are determined as follows. To find the entry in row i and column j of AB, single out row i from the matrix A and column j from the matrix B. Multiply the corresponding entries from the row and column together and then add up the resulting products.

34. Linear Algebra - Chapter 1 [YR2005] 34 Partitioned Matrices

35. Linear Algebra - Chapter 1 [YR2005] 35 Matrix Multiplication by Columns and by Rows

36. Linear Algebra - Chapter 1 [YR2005] 36 Matrix Products as Linear Combinations

37. Linear Algebra - Chapter 1 [YR2005] 37 Matrix Form of a Linear System

38. Linear Algebra - Chapter 1 [YR2005] 38 Transpose of a Matrix

39. 1.4 Inverses; Rules of Matrix Arithmetic

40. Linear Algebra - Chapter 1 [YR2005] 40 Properties of Matrix Operations ab = ba for real numbers a & b, but AB ? BA even if both AB & BA are defined and have the same size. Example:

41. Linear Algebra - Chapter 1 [YR2005] 41 Properties of Matrix Operations Theorem: Properties of A+B = B+A A+(B+C) = (A+B)+C A(BC) = (AB)C A(B+C) = AB+AC (B+C)A = BA+CA A(B-C) = AB-AC (B-C)A = BA-CA a(B+C) = aB+aC a(B-C) = aB-aC Math Arithmetic (Commutative law for addition) (Associative law for addition) (Associative for multiplication) (Left distributive law) (Right distributive law) (a+b)C = aC+bC (a-b)C = aC-bC a(bC) = (ab)C a(BC) = (aB)C

42. Linear Algebra - Chapter 1 [YR2005] 42 Properties of Matrix Operations Proof (d): Proof for both have the same size: Let size A be r x m matrix, B & C be m x n (same size). This makes A(B+C) an r x n matrix, follows that AB+AC is also an r x n matrix. Proof that corresponding entries are equal: Let A=[aij], B=[bij], C=[cij] Need to show that [A(B+C)]ij = [AB+AC]ij for all values of i and j. Use the definitions of matrix addition and matrix multiplication.

43. Linear Algebra - Chapter 1 [YR2005] 43 Properties of Matrix Operations Remark: In general, given any sum or any product of matrices, pairs of parentheses can be inserted or deleted anywhere within the expression without affecting the end result.

44. Linear Algebra - Chapter 1 [YR2005] 44 Zero Matrices A matrix, all of whose entries are zero, such as A zero matrix will be denoted by 0 or 0mxn for the mxn zero matrix. 0 for zero matrix with one column. Properties of zero matrices: A + 0 = 0 + A = A A – A = 0 0 – A = -A A0 = 0; 0A = 0

45. Linear Algebra - Chapter 1 [YR2005] 45 Identity Matrices Square matrices with 1’s on the main diagonal and 0’s off the main diagonal, such as Notation: In = n x n identity matrix. If A = m x n matrix, then: AIn = A and InA = A

46. Linear Algebra - Chapter 1 [YR2005] 46 Identity Matrices Example: Theorem: If R is the reduced row-echelon form of an n x n matrix A, then either R has a row of zeros or R is the identity matrix In.

47. Linear Algebra - Chapter 1 [YR2005] 47 Identity Matrices Definition: If A & B is a square matrix and same size ? AB = BA = I, then A is said to be invertible and B is called an inverse of A. If no such matrix B can be found, then A is said to be singular. Example:

48. Linear Algebra - Chapter 1 [YR2005] 48 Properties of Inverses Theorem: If B and C are both inverses of the matrix A, then B = C. If A is invertible, then its inverse will be denoted by the symbol A-1. The matrix is invertible if ad-bc ? 0, in which case the inverse is given by the formula

49. Linear Algebra - Chapter 1 [YR2005] 49 Properties of Inverses Theorem: If A and B are invertible matrices of the same size, then AB is invertible and (AB)-1 = B-1A-1. A product of any number of invertible matrices is invertible, and the inverse of the product is the product of the inverses in the reverse order. Example:

50. Linear Algebra - Chapter 1 [YR2005] 50 Powers of a Matrix If A is a square matrix, then we define the nonnegative integer powers of A to be A0=I An = AA...A (n>0) n factors Moreover, if A is invertible, then we define the negative integer prowers to be A-n = (A-1)n = A-1A-1...A-1 n factors Theorem: Laws of Exponents If A is a square matrix, and r and s are integers, then ArAs = Ar+s = Ars If A is an invertible matrix, then A-1 is invertible and (A-1)-1 = A An is invertible and (An)-1 = (A-1)n for n = 0, 1, 2, ... For any nonzero scalar k, the matrix kA is invertible and (kA)-1 = 1/k A-1.

51. Linear Algebra - Chapter 1 [YR2005] 51 Powers of a Matrix Example:

52. Linear Algebra - Chapter 1 [YR2005] 52 Polynomial Expressions Involving Matrices If A is a square matrix, m x m, and if is any polynomial, then we define Example:

53. Linear Algebra - Chapter 1 [YR2005] 53 Properties of the Transpose Theorem: If the sizes of the matrices are such that the stated operations can be performed, then ((A)T)T = A (A+B)T = AT + BT and (A-B)T = AT – BT (kA)T = kAT, where k is any scalar (AB)T = BTAT The transpose of a product of any number of matrices is equal to the product of their transpose in the reverse order.

54. Linear Algebra - Chapter 1 [YR2005] 54 Invertibility of a Transpose Theorem: If A is an invertible matrix, then AT is also invertible and (AT)-1 = (A-1)T Example:

55. Linear Algebra - Chapter 1 [YR2005] 55 Exercise Show that if a square matrix A satisfies A2-3A+I=0, then A-1=3I-A Let A be the matrix Determine whether A is invertible, and if so, find its inverse. [Hint. Solve AX = I by equating corresponding entries on the two sides.]

56. 1.5 Elementary Matrices and a Method for Finding A-1

57. Linear Algebra - Chapter 1 [YR2005] 57 Elementary Matrices Definition: An n x n matrix is called an elementary matrix if it can be obtained from the n x n identity matrix In by performing a single elementary row operation. Example: Multiply the second row of I2 by -3. Interchange the second and fourth rows of I4. Add 3 times the third row of I3 to the first row.

58. Linear Algebra - Chapter 1 [YR2005] 58 Elementary Matrices Theorem: (Row Operations by Matrix Multiplication) If the elementary matrix E results from performing a certain row operation on Im and if A is an m x n matrix, then the product of EA is the matrix that results when this same row operation is performed on A. Example: EA is precisely the same matrix that results when we add 3 times the first row of A to the third row.

59. Linear Algebra - Chapter 1 [YR2005] 59 Elementary Matrices If an elementary row operation is applied to an identity matrix I to produce an elementary matrix E, then there is a second row operation that, when applied to E, produces I back again. Inverse operation

60. Linear Algebra - Chapter 1 [YR2005] 60 Elementary Matrices Theorem: Every elementary matrix is invertible, and the inverse is also an elementary matrix. Theorem: (Equivalent Statements) If A is an n x n matrix, then the following statements are equivalent, that is, all true or all false. A is invertible Ax = 0 has only the trivial solution. The reduced row-echelon form of A is In. A is expressible as a product of elementary matrices.

61. Linear Algebra - Chapter 1 [YR2005] 61 Elementary Matrices Proof: Assume A is invertible and let x0 be any solution of Ax=0. Let Ax=0 be the matrix form of the system

62. Linear Algebra - Chapter 1 [YR2005] 62 Elementary Matrices Assumed that the reduced row-echelon form of A is In by a finite sequence of elementary row operations, such that: By theorem, E1,…,En are invertible. Multiplying both sides of equation on the left we obtain: This equation expresses A as a product of elementary matrices. If A is a product of elementary matrices, then the matrix A is a product of invertible matrices, and hence is invertible. Matrices that can be obtained from one another by a finite sequence of elementary row operations are said to be row equivalent. An n x n matrix A is invertible if and only if it is row equivalent to the n x n identity matrix.

63. Linear Algebra - Chapter 1 [YR2005] 63 A Method for Inverting Matrices To find the inverse of an invertible matrix, we must find a sequence of elementary row operations that reduces A to the identity and then perform this same sequence of operations on In to obtain A-1. Example: Adjoin the identity matrix to the right side of A, thereby producing a matrix of the form [A|I] Apply row operations to this matrix until the left side is reduced to I, so the final matrix will have the form [I|A-1].

64. Linear Algebra - Chapter 1 [YR2005] 64 A Method for Inverting Matrices Added –2 times the first row to the second and –1 times the first row to the third. Added 2 times the second row to the third. Multiplied the third row by –1. Added 3 times the third row to the second and –3 times the third row to the first. We added –2 times the second row to the first.

65. Linear Algebra - Chapter 1 [YR2005] 65 A Method for Inverting Matrices Often it will not be known in advance whether a given matrix is invertible. If elementary row operations are attempted on a matrix that is not invertible, then at some point in the computations a row of zeros will occur on the left side. Example: Added -2 times the first row to the second and added the first row to the third. Added the second row to the third.

66. Linear Algebra - Chapter 1 [YR2005] 66 Exercises Consider the matrices Find elementary matrices, E1, E2, E3, and E4, such that E1A=B E2B=A E3A=C E4C=A

67. Linear Algebra - Chapter 1 [YR2005] 67 Exercises Express the matrix: in the form A = E F G R, where E, F, G are elementary matrices, and R is in row-echelon form.

68. 1.6 Further Results on Systems of Equations and Invertibility

69. Linear Algebra - Chapter 1 [YR2005] 69 Linear Systems Theorem: Solving Linear Systems by Matrix Inversion: If A is an invertible n x n matrix, then for each n x 1 matrix b, the system of equations Ax = b has exactly one solution, namely, x = A-1b. Linear systems with a common coefficient matrix. Ax=b1, Ax=b2, Ax=b3, ..., Ax=bk If A is invertible, then the solutions x1=A-1b1, x2=A-1b2, x3=A-1b3, ..., xk=A-1bk This can be efficiently done using Gauss-Jordan Elimination on [A|b1|b2|...|bk]

70. Linear Algebra - Chapter 1 [YR2005] 70 Linear Systems Example: (a) (b) The solution: (a) x1=1, x2=0, x3=1 (b) x1=2, x2=1, x3=-1

71. Linear Algebra - Chapter 1 [YR2005] 71 Properties of Invertible Matrices Theorem: Let A be a square matrix. If B is a square matrix satisfying BA=I, then B=A-1. If B is a square matrix satisfying AB=I, then B=A-1. Theorem: Equivalent Statements A is invertible Ax=0 has only the trivial solutions The reduced row-echelon form of A is In A is expresssible as a product of elementary matrices Ax=b is consistent for every n x 1 matrix b Ax=b has exactly one solution for every n x 1 matrix b

72. Linear Algebra - Chapter 1 [YR2005] 72 Properties of Invertible Matrices Theorem: Let A and B be square matrices of the same size. If AB is invertible, then A and B must also be invertible. A fundamental problem. Let A be a fixed m x n matrix. Find all m x 1 matrices b such that the system of equations Ax=b is consistent.

73. Linear Algebra - Chapter 1 [YR2005] 73 Exercises Solve the system by inverting the coefficient matrix. Find condition that b’s must satisfy for the system to be consistent.

74. 1.7 Diagonal, Triangular, and Symmetric Matrices

75. Linear Algebra - Chapter 1 [YR2005] 75 Diagonal Matrices A square matrix in which all the entries off the main diagonal are zero. Example: A diagonal matrix is invertible if and only if all of its diagonal entries are nonzero.

76. Linear Algebra - Chapter 1 [YR2005] 76 Diagonal Matrices Example:

77. Linear Algebra - Chapter 1 [YR2005] 77 Triangular Matrices Lower triangular = a square matrix in which all the entries above the main diagonal are zero. Upper triangular = a square matrix in which all the entries under the main diagonal are zero. Triangular = a matrix that is either upper triangular or lower triangular.

78. Linear Algebra - Chapter 1 [YR2005] 78 Triangular Matrices Theorem: (basic properties of triangular matrices) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. The product of lower triangular matrices is lower triangular, and the product of upper triangular matrices is upper triangular. A triangular matrix is invertible if and only its diagonal entries are all nonzero. The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular.

79. Linear Algebra - Chapter 1 [YR2005] 79 Triangular Matrices Example: The matrix A is invertible, since its diagonal entries are nonzero, but the matrix B is not. This inverse is upper triangular. This product is upper triangular.

80. Linear Algebra - Chapter 1 [YR2005] 80 Symmetric Matrices A square matrix A is called symmetric if A = AT. A matrix A = [aij] is symmetric if and only if aij=aji for all values of i and j.

81. Linear Algebra - Chapter 1 [YR2005] 81 Symmetric Matrices Theorem: If A and B are symmetric matrices with the same size, and if k is any scalar, then AT is symmetric A+B and A-B are symmetric kA is symmetric Theorem: If A is an invertible matrix, then A-1 is symmetric. If A is an invertible matrix, then AAT and ATA are also invertible.

82. Linear Algebra - Chapter 1 [YR2005] 82 Exercise Find all values of a, b, and c for which A is symmetric. Find all values of a and b for which A and B are both not invertible.