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Systems of Linear Equations and Matrices

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Title: Systems of Linear Equations and Matrices


1
Linear Algebra
  • Lecturers
  • Heru Suhartanto, PhD, heru_at_cs.ui.ac.id
  • Yova Ruldeviyani, MKom, yovayg_at_gmail.com
  • Schedule (at C3111)
  • Tuesday, 1.00 2.40 PM
  • Thursday, 1.00 1.50 PM
  • Marking scheme
  • 2 exams (mid 15, final 20) reqs attendance
    75
  • 4 quizes, 40
  • 5 assignments, 25
  • Reference
  • Horward Anton, Elementary Linear Algebra, 8-th
    Ed, John Wiley Sons, Inc, 2000
  • More information (will be updated soon) at
  • http//telaga.cs.ui.ac.id/WebKuliah/LinearAlgebra0
    5/

2
Systems of Linear Equation and Matrices
  • CHAPTER 1
  • FASILKOM UI 05

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 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 Equations
  • The equations are not linear.
  • A solution of
    is a sequence of n numbers s1, s2, ..., sn ?
    they satisfy the equation when x1s1, x2s2, ...,
    xnsn (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 Systems / System of Linear Equations
  • Is A finite set of linear equations in the vars
    x1, ..., xn
  • s1, ..., sn is called a solution if x1s1, ...,
    xnsn is a solution of every equation in the
    system.
  • Ex.
  • x11, x22, x3-1 the solution
  • x11, x28, x31 is not, satisfy only the first
    eq.
  • System that has no solution inconsistent
  • System that has at least one solution consistent
  • Consider

8
Linear Systems
  • (x,y) lies on a line if and only if the numbers x
    and y satisfy the equation of the line. Solution
    points of intersection l1 l2
  • l1 and l2 may be parallel
  • no intersection, no solution
  • l1 and l2 may intersect
  • at only one point one solution
  • l1 and l2 may coincide
  • infinite many points of intersection,
  • infinitely many solutions

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, as and bs are constants
  • aij, i indicates the equation in which the
    coefficient occurs and j indicates which unknown
    it multiplies

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
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
Elementary Row Operations (Example)
  • r2 -2r1 r2
  • r3 -3r1 r3

13
Elementary Row Operations (Example)
  • r2 ½ r2
  • r3 -3r2 r3
  • r3 -2r3

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
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
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
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
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
Elimination Methods
  • Step 6 add suitable multiplies of each row to
    the rows above to introduce zeros above the
    leading 1s.
  • r2 7/2 r3 r2
  • r1 -6r3 r1
  • r1 5r2 r1

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
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
Back-Substitution
  • Step 3. Assign arbitrary values to the free
    variables parameters, if any

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
    x10,x20,...,xn0 as a solution trivial
    solution. Other solutions are called nontrivial
    solutions.

25
Homogeneous Linear Systems
  • Example Gauss-Jordan Elimination

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 st0

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
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
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
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 AB if and only if (A)ij(B)ij, or
    equivalently aijbij for all i and j.
  • Definition
  • If A and B are matrices of the same size, then
    the sum AB is the matrix obtained by adding the
    entries of B to the corresponding entries of A,
    and the difference AB 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
Operations on Matrices
  • If A aij and B bij have the same size,
    then
  • (AB)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
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
Partitioned Matrices
35
Matrix Multiplication by Columns and by Rows
36
Matrix Products as Linear Combinations
37
Matrix Form of a Linear System
38
Transpose of a Matrix
39
  • 1.4 Inverses Rules of Matrix Arithmetic

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
Properties of Matrix Operations
  • Theorem Properties of
  • AB BA
  • A(BC) (AB)C
  • A(BC) (AB)C
  • A(BC) ABAC
  • (BC)A BACA
  • A(B-C) AB-AC
  • (B-C)A BA-CA
  • a(BC) aBaC
  • a(B-C) aB-aC
  • Math Arithmetic
  • (Commutative law for addition)
  • (Associative law for addition)
  • (Associative for multiplication)
  • (Left distributive law)
  • (Right distributive law)
  • (ab)C aCbC
  • (a-b)C aC-bC
  • a(bC) (ab)C
  • a(BC) (aB)C

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(BC) an r x n matrix, follows that
    ABAC is also an r x n matrix.
  • Proof that corresponding entries are equal
  • Let Aaij, Bbij, Ccij
  • Need to show that A(BC)ij ABACij for all
    values of i and j.
  • Use the definitions of matrix addition and matrix
    multiplication.

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
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
Identity Matrices
  • Square matrices with 1s on the main diagonal and
    0s 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
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
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
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
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
Powers of a Matrix
  • If A is a square matrix, then we define the
    nonnegative integer powers of A to be
  • A0I An AA...A (ngt0)
  • 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 Ars 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
Powers of a Matrix
  • Example

52
Polynomial Expressions Involving Matrices
  • If A is a square matrix, m x m, and if
  • is any polynomial, then we define
  • Example

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
  • (AB)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
Invertibility of a Transpose
  • Theorem If A is an invertible matrix, then AT is
    also invertible and (AT)-1 (A-1)T
  • Example

55
Exercise
  • Show that if a square matrix A satisfies
    A2-3AI0, then A-13I-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
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
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
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
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
Elementary Matrices
  • Proof
  • Assume A is invertible and let x0 be any
    solution of Ax0.
  • Let Ax0 be the matrix form of the system

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
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 AI
  • Apply row operations to this matrix until the
    left side is reduced to I, so the final matrix
    will have the form IA-1.

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
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
Exercises
  • Consider the matrices
  • Find elementary matrices, E1, E2, E3, and E4,
    such that
  • E1AB
  • E2BA
  • E3AC
  • E4CA

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 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.
  • Axb1, Axb2, Axb3, ..., Axbk
  • If A is invertible, then the solutions
  • x1A-1b1, x2A-1b2, x3A-1b3, ..., xkA-1bk
  • This can be efficiently done using Gauss-Jordan
    Elimination on Ab1b2...bk

70
Linear Systems
  • Example (a) (b)
  • The solution
  • (a) x11, x20, x31
  • (b) x12, x21, x3-1

71
Properties of Invertible Matrices
  • Theorem Let A be a square matrix.
  • If B is a square matrix satisfying BAI, then
    BA-1.
  • If B is a square matrix satisfying ABI, then
    BA-1.
  • Theorem Equivalent Statements
  • A is invertible
  • Ax0 has only the trivial solutions
  • The reduced row-echelon form of A is In
  • A is expresssible as a product of elementary
    matrices
  • Axb is consistent for every n x 1 matrix b
  • Axb has exactly one solution for every n x 1
    matrix b

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 Axb
    is consistent.

73
Exercises
  • Solve the system by inverting the coefficient
    matrix.
  • Find condition that bs must satisfy for the
    system to be consistent.

74
  • 1.7 Diagonal, Triangular,
  • and Symmetric Matrices

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
Diagonal Matrices
  • Example

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
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
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
Symmetric Matrices
  • A square matrix A is called symmetric if A AT.
  • A matrix A aij is symmetric if and only if
    aijaji for all values of i and j.

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
  • AB 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
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.
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