??????13 : Solutions of Linear Systems

1
??????13 Solutions of Linear Systems

• ??????? (Kuang-Chi Chen)
• chichen6_at_mail.tcu.edu.tw

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Linear Equations and Matrices Solutions of
Linear Systems of Equations
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Solutions of Linear Systems of Equations

• 1.6 Solutions of Linear Systems of Equations

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Row Echelon Form

• Definition Row echelon form (r.e.f.)
• An m?n matrix A is said to be in row echelon form
  if
• (a) All zero rows, if there are any, appear at
  the bottom of the matrix
• (b) The first nonzero entry from the left of a
  nonzero row is a 1 a leading one of the row
• (c) For each nonzero row, the leading one appears
  to the right and below any leading ones in
  preceding rows

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Reduced Row Echelon Form

• Definition Reduced row echelon form
• An m?n matrix A is said to be in reduced row
  echelon form if
• (a) A is in row echelon form
• (b) If a column contains a leading one, then all
  other entries in that column are zero
• (???? ???????)

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Example 1 - in row echelon form

• E.g. 1

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Example 2 reduced row echelon form

• E.g. 2

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E.g. 2 not reduced row echelon form

• E.g. 2

,
,
,
Nonzero element above leading 1 in row 2
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Three Basic Types of Elementary Row Operations

• Type 1 Interchange
• row i and row j are interchanged
• Type 2 Multiply
• row i row i times c
• Type 3 Add
• Add d times row r of A to row s of A
• row s row s d ? row r

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Example 3

• E.g. 3

E1 ?
? E2
E3
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Row Equivalence

• Definition Row Equivalence
• An m?n matrix A is said to be row equivalence
  to an m?n matrix B if B can be obtained by
  applying a finite sequence of elementary row
  operations to the matrix A .

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Example 4

• E.g. 4

E3 ?
E1
E2 ?
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Theorem 1.5

• Theorem 1.5
• Every m?n matrix is row equivalent to a matrix
  in row echelon form .

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E.g. 5 - Procedure of Row Echelon Form

• E.g. 5
• Step 1 Find the pivotal column
• Step 2 Identify the pivot in the pivotal column

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(contd)

• E.g. 5
• Step 3 Interchange if necessary so that the
  pivot is in the 1st row
• Step 4 Multiply so that the pivot equals to 1

pivot
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(contd)

• E.g. 5
• Step 5 Make all entries in the pivot column,
  except the entry where the pivot was located,
  equal to zero

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(contd)

• E.g. 5
• Step 6 Ignore the first row and repeat
• ?
• ? ?

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Example 6

• Example 6

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Remark

• Remark
• - There may be more than one matrix in row
  echelon form that is row equivalent to a given
  matrix.
• - A matrix in row echelon form (r.e.f.) that is
  row equivalent to A is called
• a row echelon form of A.

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Theorem 1.6

• Theorem 1.6
• - Every m?n matrix is row equivalent to a
  unique matrix in reduced row echelon form.

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Example 7 r.e.f. to reduced r.e.f.

• E.g. 7

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Theorem 1.7

• Theorem 1.7
• Let Ax b and Cx d be two linear systems
  each of m equations in n unknowns. If the
  augmented matrices Ab and Cd of these
  systems are row equivalent, then both linear
  systems have the same solutions.

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Corollary 1.1

• Corollary 1.1
• If A and C are row equivalent m?n matrices,
  then the linear system Ax 0 and Cx 0 have
  exactly the same solutions.

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Gauss-Jordan Reduction Procedure

• The Gauss-Jordan reduction procedure
• Step 1. Form the augmented matrix Ab
• Step 2. Obtain the reduced row echelon form
  Cd of the augmented matrix Ab
  by using elementary row operations
• Step 3. For each nonzero row of Cd, solve the
  corresponding equation.
• (augmented matrix ????)

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Gauss Elimination Procedure

• The Gauss elimination procedure
• Step 1. Form the augmented matrix Ab
• Step 2. Obtain a row echelon form Cd of
  the augmented matrix Ab by using
  elementary row operations
• Step 3. Solving the linear system corresponding
  to Cd, by back substitution (????).

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Example 8

• E.g. 8
• Solve the linear system by Gauss-Jordan
  reduction
• - Step 1

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(contd)

• E.g. 8 - Solve the linear system by Gauss-Jordan
  reduction
• - Step 2

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(contd)

• E.g. 8 - Solve the linear system by Gauss-Jordan
  reduction
• - Step 3 x 2
• y -1
• z 3

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Example 9

• Example 9
• - Solve the linear system by Gauss-Jordan
  reduction
• x y 2z 5w 3
• 2x 5y z 9w -3
• 2x y z 3w -11
• x 3y 2z 7w -5

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(contd)

• Example 9
• - Step 1
• - Step 2

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(contd)

• E.g. 9 - Step 3
• 
  leading variables
• 
  a free variable

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Example 10

• Example 10
• - Solve the linear system by Gauss-Jordan
  reduction
• x1 2x2 3x4 x5 2
• x1 2x2 x3 3x4 x5 2x6 3
• x1 2x2 3x4 2x5 x6 4
• 3x1 6x2 x3 9x4 4x5 3x6 9

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(contd)

• Example 10
• - Step 1
• - Step 2

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(contd)

• Example 10 - Step 3
• 
  leading variables
• 
  free variables

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Example 11

• Example 11
• - Solve the linear system by Gauss elimination
• x 2y 3z 9
• 2x y z 8
• 3x z 3

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(contd)

• Example 11
• - Step 1
• - Step 2

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(contd)

• Example 11 - Step 3
• - By back substitution

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Example 12

• Example 12
• - Solve the linear system by Gauss elimination
• x 2y 3z 4w 5
• x 3y 5z 7w 11
• x z w -6

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(contd)

• Example 12
• - Step 1
• - Step 2
• - Step 3 ? 0x 0y 0z 0w 1 ? No
  solutions !!

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Consistent and Inconsistent

• Consistent and inconsistent
• - Consistent Linear systems with at least one
  solution
• - Inconsistent Linear systems with no solutions

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Homogeneous Systems

• A system of linear equations is said to be
  homogeneous if all the constant terms are zeros.
• a11x1 a12x2 a1nxn 0
• a21x1 a22x2 a2nxn 0
• am1x1 am2x2 amnxn 0
• ? Ax 0
• Thus, a homogeneous system always has the
  solution x1 x2 xn 0 ? the trivial
  solution

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Example 13

• Example 13

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Example 14

• Example 14

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Theorem 1.8

• Theorem 1.8
• A homogeneous system of m equations in n
  unknowns has a non-trivial solution if m lt n,
  that is, if the number of unknowns exceeds the
  number of equations.
• namely, a homogeneous system has more
  variables than equations has many solutions.
• (a homogeneous system???? non-trivial
  solution ???)

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Example 15 - A Homogeneous System

• E.g. 15

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(contd)

• If let

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A Homogeneous System Example
x xp xh xp is a particular solution
to the given system Axp b , where b
3 -3 -11 -5T xh is a solution to the
associated homogeneous system Axh 0 .
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Polynomial Interpolation

• Polynomial Interpolation
• - The general form
• y an 1xn 1 an 2xn 2 a1x
  a0
• E.g. n 3, y a2x2 a1x a0
• Given three distinct points (x1 , y1), (x2 , y2),
  (x3 , y3),
• we have
• y1 a2x12 a1x1 a0
• y2 a2x22 a1x2 a0
• y3 a2x32 a1x3 a0

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Example 16

• Example 16 - Find the quadratic polynomial that
  interpolates the points (1, 3), (2, 4), (3, 7)

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Example 17 Temperature Distribution

• T1 (260 100 T2 T3 )/4 or 4T1 T2
  T3 160
• T2 (T1 100 40 T4 )/4 or -T1 4T2
  T4 140
• T3 (60 T1 T4 0)/4 or -T1 4T3
  T4 60
• T4 (T2 T3 40 0)/4 or -T2 T3
  4T4 40
• ?
• ? T1 65, T2 60, T3 40, T4 35 .

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Linear Equations and Matrices The Inverse of A
Matrix
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The Inverse of A Matrix

• 1.7 The inverse of a matrix
• Definition
• - An n?n matrix A is called nonsingular (or
  invertible ???) if there exists an n?n matrix B
  such that AB BA In .
• - The matrix B is called the inverse of A
• - If there exists no such matrix B, then A is
  called singular (or noninvertible)
• - A is also an inverse of B

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Example 1

• Example 1
• ? AB BA I2
• - B is an inverse of A and A is nonsingular.

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Theorem 1.9

• Theorem 1.9
• An inverse of a matrix, if exists, is unique.
• (proof)
• Let B and C be inverses of A.
• Then AB BA In, and AC CA In.
• Thus, C(AB) CIn
• (CA)B C
• InB C , i.e., B C .

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Example 2 - Find the Inverse
For the matrix A, find the inverse If exists,
let the inverse A-1 be such that
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(contd)
and
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Example 3

• Example 3

and
? No solution singular
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Theorem 1.10

• Thm. 1.10 - Properties of an inverse
• - If A is nonsingular, then A-1 is nonsingular
• and (A-1)-1 A
• - If A and B are nonsingular matrices, then AB
  is nonsingular and (AB)-1 B-1 A-1
• - If A is a nonsingular matrix, then (AT)-1
  (A-1)T .

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Example 4

• Example 4

? and
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Corollary1.2

• Corollary 1.2
• - If A1 , A2 , , Ar are n?n nonsingular
  matrices, then (A1 A2 Ar) is nonsingular and
  (A1 A2 Ar)-1 Ar-1 A2-1 A1-1 .

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Theorem 1.11

• Theorem 1.11
• Suppose that A, B are n?n matrices,
• - If AB In , then BA In
• - If BA In , then AB In .

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The Way to Find A-1

• A practical method for finding A-1
• Step 1. Form the 2?2n matrix A In obtained by
  adjoining the identity matrix In to the given
  matrix A
• Step 2. Compute the reduced row echelon form of
  the matrix obtained in Step 1 by using elementary
  row operations
• Step 3. Suppose that Step 2 has produced the
  matrix C D in reduced row echelon form
• If C In , then D A-1
• If C ? In , then C has a row of zeros and the
  matrix A is singular .

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Example 5 Find the Inverse

• E.g. 5 Find the inverse

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Example 6 Find the Inverse

• E.g. 6 - Find the inverse

The left-half matrix cannot have a one in the (3,
3) location, the reduced echelon form cannot be
I3. Thus A-1 does not exist.
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Theorem 1.12 1.13

• Theorem 1.12
• An n?n matrix is nonsingular iff it is row
  equivalence to In .
• Theorem 1.13
• If A is an n?n matrix, the homogeneous system
  Ax 0 has a nontrivial solution iff
• A is singular.

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Proof of Theorem 1.13

• Proof of Theorem 1.13
• Suppose that A is nonsingular, then A-1 exists
  and
• A-1(Ax) A-1 0
• (A-1A)x 0
• In x 0
• x 0 ? Ax 0 has a trivial
  solution
• (contradiction to a non-trivial solution,
  hence A must be singular)

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Example 8

• Example 8
• Consider the homogeneous system Ax 0, where
  A is the matrix (A is nonsingular)

Gauss-Jordan reduction
The trivial solution x 0
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Example 9

• Example 9
• - Consider the homogeneous system Ax 0, where
  A is the matrix (A is singular)

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Theorem 1.14

• Theorem 1.14
• If A is an n?n matrix, then A is nonsingular
  iff the linear system Ax b has a unique
  solution for every n?1 matrix b .

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Summary The Symmetry, Singularity, Inverse of A
Matrix
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Some Special Matrix

• 4. A square matrix A is said to be antisymmetric
  if -AT A. (i) If A is square, prove that A AT
  is symmetric and A AT is antisymmetric
• (ii) any square matrix A can be decomposed
  into the sum of a symmetric matrix B and an
  antisymmetric matrix C A B C .
• 5. Given two symmetric matrices of the same
  size, A and B, then a necessary and sufficient
  condition for the product AB to be symmetric is
  that AB BA.

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Some Special Matrix

• 1. A square matrix A is said to be normal if AAT
  ATA. All symmetric matrices are normal
• 2. A square matrix A is said to be idempotent if
  A2 A. If A is idempotent then AT is also
  idempotent
• 3. A square matrix A is said to nilpotent if
  there is a positive integer p such that Ap O.
  The least integer such that Ap O is called the
  degree of nilpotency of the matrix. If A is
  nilpotent, then AT is also nilpotent with the
  same degree of nilpotency.

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List of Nonsingular Equivalences

• Nonsingular equivalences
• 1. A is nonsingular
• 2. x 0 is the only solution to Ax 0
• 3. A is row equivalence to In
• 4. The linear system Ax b has a unique solution
  for every n?1matrix b .

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Properties of Matrix Inverse

• Properties of Matrix Inverse
• 1. (A-1)-1 A
• 2. (cA)-1 (1/c)A-1 , where c is a nonzero
  scalar
• 3. (AB)-1 B-1A-1
• 4. (An)-1 (A-1)n
• 5. (AT)-1 (A-1)T , where T transpose.

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Conditions of Matrix Inverse

• A matrix has no inverse, if
• (i) two rows are equal
• (ii) two columns are equal (Use the transpose)
• (iii) it has a column of zeros.

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The Inverse of 2?2 Matrix

• If A , show that A-1
  .
• Note The cancellation law doesnt hold.
• That is, AB AC doesnt imply that B C .
• Also, AB O doesnt imply that A O or B O.
• However, if A is an invertible matrix, then
• if AB AC , then B C
• if AB 0, then B 0 .