systems of linear equation and matrices n.
Download
Skip this Video
Download Presentation
Systems of Linear Equation and Matrices

Loading in 2 Seconds...

play fullscreen
1 / 50

Systems of Linear Equation and Matrices - PowerPoint PPT Presentation


  • 94 Views
  • Uploaded on

Systems of Linear Equation and Matrices. CHAPTER 1 FASILKOM UI 05. YR. 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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Systems of Linear Equation and Matrices' - kyna


Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
introduction matrices
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
introduction to systems of linear equations
Introduction to Systems

of Linear Equations

linear equations
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).
linear equations1
Linear Equations
    • The equations are not linear.
    • A solution of is a sequence of n numbers s1, s2, ..., snЭ 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
linear systems system of linear equations
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 x1=s1, ..., xn=sn is a solution of every equation in the system.
  • Ex.
    • x1=1, x2=2, x3=-1 the solution
    • x1=1, x2=8, x3=1 is not, satisfy only the first eq.
    • System that has no solution : inconsistent
    • System that has at least one solution: consistent
  • Consider:
linear systems
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

linear systems1
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
augmented matrices
Augmented Matrices
  • Example:
  • Remark: when constructing, the unknowns must be written in the same order in each equation and the constants must be on right.
augmented matrices1
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.
elementary row operations example
Elementary Row Operations (Example)
  • r2= -2r1 + r2
  • r3 = -3r1 + r3
elementary row operations example1
Elementary Row Operations (Example)
  • r2 = ½ r2
  • r3 = -3r2 + r3
  • r3 = -2r3
elementary row operations example2
Elementary Row Operations (Example)
  • r1 = r1 – r2
  • r1 = -11/2 r3 + r1
  • r2 = 7/2 r3 + r2
  • Solution:
echelon forms
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.
echelon forms1
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:
elimination methods
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.
elimination methods1
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
elimination methods2
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
elimination methods3
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.
back substitution
Back-substitution
  • Bring the augmented matrix into row-echelon form only and then solve the corresponding system of equations by back-substitution.
  • Example:
properties of matrix operations
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:
properties of matrix operations1
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

Properties of Matrix Operations
properties of matrix operations2
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.
properties of matrix operations3
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.
zero matrices
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. 0for zero matrix with one column.
  • Properties of zero matrices:
    • A + 0 = 0 + A = A
    • A – A = 0
    • 0 – A = -A
    • A0 = 0; 0A = 0
identity matrices
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
identity matrices1
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.
identity matrices2
Identity Matrices
  • Definition: If A & B is a square matrix and same size Э 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:
properties of inverses
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

properties of inverses1
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:
powers of a matrix
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.
polynomial expressions involving matrices
Polynomial Expressions Involving Matrices
  • If A is a square matrix, m x m, and if

is any polynomial, then we define

  • Example:
properties of the transpose
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.
invertibility of a transpose
Invertibility of a Transpose
  • Theorem: If A is an invertible matrix, then AT is also invertible and (AT)-1 = (A-1)T
  • Example:
elementary matrices
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.
elementary matrices1
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.
elementary matrices2
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
elementary matrices3
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.
elementary matrices4
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

elementary matrices5
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.
a method for inverting matrices
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].
a method for inverting matrices1
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.

a method for inverting matrices2
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:
ad