Row Reduction

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Row Reduction Echelon FormsBasic Definitions
01/24/2008

• A matrix is in (Row) Echelon Form (REF) if
• All nonzero rows are above any rows of all zeros.
• Each leading (nonzero) entry of a row is in a
  column to the right of the leading entry of the
  row above it.
• All entries in a column below a leading entry are
  zero.

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Row Reduction Echelon FormsBasic Definitions

• A matrix is in (Row) Echelon Form (REF) if
• All nonzero rows are above any rows of all zeros.
• Each leading (nonzero) entry of a row is in a
  column to the right of the leading entry of the
  row above it.
• All entries in a column below a leading entry are
  zero.

• A matrix is in Reduced (Row) Echelon Form (RREF)
• if it is in echelon form and in addition
• The leading entry in each nonzero row is 1.
• Each leading 1 is the only nonzero entry in its
  column.

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Row Reduction Echelon FormsBasic Definitions

• (Row) Echelon Form
• Reduced (Row) Echelon Form

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Row Reduction Echelon FormsBasic Definitions

• A matrix is in (Row) Echelon Form (REF) if
• All nonzero rows are above any rows of all zeros.
• Each leading (nonzero) entry of a row is in a
  column to the right of the leading entry of the
  row above it.
• All entries in a column below a leading entry are
  zero.
• A matrix is in Reduced (Row) Echelon Form (RREF)
• if it is in echelon form and in addition
• The leading entry in each nonzero row is 1.
• Each leading 1 is the only nonzero entry in its
  column.
• Uniqueness Each matrix is row equivalent to one
• and only one reduced row echelon matrix.

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Row Reduction Echelon FormsWhich are in REF?
In RREF?
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Row Reduction Echelon FormsBasic Definitions

• A matrix is in (Row) Echelon Form (REF) if
• All nonzero rows are above any rows of all zeros.
• Each leading (nonzero) entry of a row is in a
  column to the right of the leading entry of the
  row above it.
• All entries in a column below a leading entry are
  zero.
• A Pivot Position in a matrix A is a location in A
  that
• corresponds to a leading entry in an echelon form
  of A.
• A Pivot Column is a column containing a pivot
  position.

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Row Reduction Echelon FormsBasic Definitions

• A Pivot Position in a matrix A is a location in A
  that
• corresponds to a leading entry in an echelon form
  of A.
• A Pivot Column is a column containing a pivot
  position.
• The variables in a linear system are classified
  as
• Basic Variables if they correspond to pivot
  columns
• Free Variables if they correspond to non-pivot
  columns of the coefficient matrix

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Row Reduction Echelon FormsBasic Definitions

• A Pivot Position in a matrix A is a location in
  A that corresponds to a leading entry in an
  echelon form of A.A Pivot Column is a column
  containing a pivot position.
• The variables in a linear system are classified
  as
• Basic Variables if they correspond to pivot
  columns
• Free Variables if they correspond to non-pivot
  columns of the coefficient matrix
• The General Solution of a linear system is
  obtained by
• Finding the RREF of its augmented matrix
• Solving each equation in the associated system
  for its leading (basic) variable in terms of the
  free variables
• Stating which variables are the free variables.

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Row Reduction Echelon FormsExample (text, page
21)

• The variables in a linear system are classified
  as
• Basic Variables if they correspond to pivot
  columns
• Free Variables if they correspond to non-pivot
  columns of the coefficient matrix
• Find the general solution of the linear system
  whose augmented matrix has been reduced to
• Basic Variables
• Free Variables

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Row Reduction Echelon FormsExample

• The General Solution of a linear system is
  obtained by
• Finding the RREF of its augmented matrix
• Solving each equation in the associated system
  for its leading (basic) variable in terms of the
  free variables
• Stating which variables are the free variables.
• Basic Variables x1, x3, x5
• Free Variables x2, x4

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Row Reduction Echelon FormsExample

• The General Solution of a linear system is
  obtained by
• Finding the RREF of its augmented matrix
• Note that the symbols separating the matrices are
  tildes (meaning is equivalent to) and NOT
  equal signs (meaning is the same matrix as)

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Row Reduction Echelon FormsExample

• The General Solution of a linear system is
  obtained by
• Solving each equation in the associated system
  for its leading (basic) variable in terms of the
  free variables
• Basic Variables x1, x3, x5
• Free Variables x2, x4
• Associated System

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Row Reduction Echelon FormsExample

• The General Solution of a linear system is
  obtained by
• Solving each equation in the associated system
  for its leading (basic) variable in terms of the
  free variables
• Associated System General or Parametric
  Solution

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Existence and Uniqueness Questions

• How can we tell from the augmented matrix of a
  system
• Existence If there are any solutions?
• Uniqueness If there is a solution, is it unique
  or are there infinitely many solutions?
• A linear system is consistent if and only if
• The rightmost column of the augmented matrix is
  not a pivot column.
• The echelon form of the augmented matrix has no
  row of the form
• 0 0 . . . 0 b with b nonzero.
• If a linear system is consistent, the solution
  set contains
• A unique solution when there are no free
  variables
• Infinitely many solutions when there is at least
  one free variable

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