Systems of Linear Equations, Direct Methods

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Lecture 8

• Systems of Linear Equations, Direct Methods

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Lecture 8 Objectives

• Geometrically solve linear systems of eqns in 2
  vars
• Get the augmented matrix of a linear system of
  eqns
• Decide if a matrix is in (reduced) row echelon
  form
• Use elementary operations to reduce a matrix to
  (reduced) row echelon form
• Algebraically solve linear systems of equations
  using
• Gaussian elimination
• Gauss-Jordan elimination
• Find the value(s) of parameter(s) that make a
  linear system have
• a unique solution
• infinitely many solutions
• no solution

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Example

• Solve the following equations
• x 3y ?1
• 3x 2y 4
• Note 1 Geometrically you are looking for the
  intersection of the two lines in the x-y plane.
  Here we get a unique solution.

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Example

• Solve the following equations
• x 3y ?1
• 3x 2y 4
• Note 2 These two equations can be written as

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Example

• Solve the following equations
• x 3y ?1
• 3x 9y 4
• Note These are two different parallel lines.
  Thus, we get no solution (the solution set is
  empty). We call this system inconsistent.

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Example

• Solve the following equations
• x 3y 1
• 3x 9y 3
• Note These are two identical lines. Thus, we get
  infinitely many solutions, namely y t (any
  arbitrary value), and x 1 ? 3t. Thus, the
  solutions can be written as (x, y) (1 ? 3t,
  t), where t?R.

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Example

• Solve the matrix equation

8
Example

• I.e., solve the following equations
• x 2y ? 3z ?4
• 2x ? y 2z 10
• 3x y ? 4z 3
• Note Geometrically you are looking for the
  intersection of three planes in space.

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Solution Step 1 Elimination

• x 2y ? 3z ?4 (1)
• 2x ? y 2z 10 (2)
• 3x y ? 4z 3 (3)
• We use Eq. 1 to eliminate x from Eqs. 2, 3
• x 2y ? 3z ?4 (1)
• ?5y 8z 18 (2)
• ?5y 5z 15 (3)
• Note Those 3 equations are equivalent to the
  original set of equations, i.e. they have the
  same set of solutions.

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Solution Step 2 Elimination

• x 2y ? 3z ?4 (1)
• ?5y 8z 18 (2)
• ?5y 5z 15 (3)
• We now use Eq. 2 to eliminate y from Eq. 3
• x 2y ? 3z ?4 (1)
• ?5y 8z 18 (2)
• ?3z ?3 (3)
• Note Again, we get 3 equivalent equations, which
  can now be easily solved.

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Solution Step 3 Back Substitution

• x 2y ? 3z ?4 (1)
• ?5y 8z 18 (2)
• ?3z ?3 (3)
• We first get z, then y, then x
• z 1
• ?5y 8(1) 18, so y ?2
• x 2(?2) ? 3(1) ?4, so x 3
• Note Thus, we finally get the unique solution
  (point, or vector) (x, y, z) (3, ?2, 1).

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Example (revisited)

• Solve the following equations
• x 2y ? 3z ?4
• 2x ? y 2z 10
• 3x y ? 4z 3
• A Simplified Method We only keep track of the
  coefficients of x, y, and z, as well as the right
  hand sides.

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Solution Step 1 Form the Augmented Matrix
(Table)

• x 2y ? 3z ?4 (1)
• 2x ? y 2z 10 (2)
• 3x y ? 4z 3 (3)
• The equations are summarized in the augmented
  matrix

Notes Each row represents an equation. Also
Ab is the augmented matrix for the system Ax
b.
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The Elementary Row Operations

• Those are operations performed on the augmented
  matrix Ab, maintaining the equivalence between
  the old and the new set of equations.
• They are
• 1) Add a multiple of a row to another
• 2) Multiply a row by a nonzero constant
• 3) Interchange two rows
• Note These operations change the augmented
  matrix, but do NOT change the set of solutions.

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Solution Step 2 Perform the legal Elementary Row
Operations
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Solution Step 3 We can stop and solve, or
continue to simplify
Thus, x 3 y ?2 z 1
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In general We can solve any system (set) of m
linear equations in n unknowns

• a11x1 a12x2 ? a1nxn b1
• a21x1 a22x2 ? a2nxn b2
• ?
• am1x1 am2x2 ? amnxn bm
• i.e. Ax b, by first getting the augmented
  matrix

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Example

• Find the solution of the system, whose augmented
  matrix is

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Solution

• We perform row operations to get the reduced row
  echelon form

Thus x4 ?5 x3 2 x2 t x1 ?11 ? 2t
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Gaussian Elimination

• Here we perform the elementary row operations
• to find an equivalent matrix in row echelon form,
  i.e. with the following properties
• 1) All zero rows are at the bottom.
• 2) The first nonzero entry of a nonzero row is
  called a pivot. Each pivot lies to the left of
  any pivot below it.
• E.g.

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Gauss-Jordan Elimination

• Here we go further to get an equivalent matrix in
  reduced row echelon form, i.e. with the following
  properties
• 1) All zero rows are at the bottom.
• 2) Each pivot lies to the left of pivots below
  it.
• 3) Each pivot must 1, and is the only nonzero
  entry in its column.
• E.g.

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The Rank of a Matrix A

• rank(A) is defined as the number of pivots (or
  nonzero rows) in the (reduced) row echelon form
  of A.
• E.g.

has rank 3, since it is row-equivalent to
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The Rank Theorem

• Let Ab be the augmented matrix of a linear
  system of m equations in n variables. Then
• 1) The system is inconsistent iff rank(A) lt
  rank(Ab)
• 2) The system has a unique solution
  iff rank(A) rank(Ab) n
• 3) The system has infinitely many solutions iff
  rank(A) rank(Ab) lt n, and
• Number of free variables n ? rank(A)

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Example

• For what values of k does the system
• x ? 2y 3z 2
• x y z k
• 2x ? y 4z k2 have
• a) No Solution
• b) A unique solution
• c) Infinitely many solutions

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• Thank you for listening.
• Wafik