Gauss Elimination

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Gauss Elimination
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A System of Linear Equations
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Two Equations, Two Unknowns Lines in a Plane
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Three Possible Types of Solutions

• 1. No solution

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Three Possible Types of Solutions

• 1. A unique solution

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Three Possible Types of Solutions

• 1. Infinitely many solutions

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Three Equations, Three UnknownsPlanes in Space
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Intesections of Planes

• What type of solution sets are represented?

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Solve the System
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Elementary Operations

• Interchange the order in which the equations are
  listed.
• Multiply any equation by a nonzero number.
• Replace any equation with itself added to a
  multiple of another equation.

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Augmented Matrix
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Row Operations

• Switch two rows.
• Multiply any row by a nonzero number.
• Replace any row by a multiple of another row
  added to it.

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Solve the System
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Echelon Form

• A rectangular matrix is in echelon form if it
  has the following properties
• 1. All nonzero rows are above any rows of all
  zeros.
• 2. Each leading entry of a row is in a column
  to the right of the leading entry of the row
  above it.

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Echelon Form
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Echelon Form
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Pivot Positions and Pivot Columns

• The positions of the first nonzero entry in each
  row are called the pivot positions.
• The columns containing a pivot position are
  called the pivot columns.

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Types of Solutions

• 1. No solution the augmented column is a
  pivot column.
• 2. A unique solution every column except
  the augmented column is a pivot column.
• 3. An infinite number of solutions some
  column of the coefficient matrix is not a pivot
  column.
• The variables corresponding to the columns that
  are not pivot columns are assigned parameters.
  These variables are called the free variables.
  The other variables may be solved in terms of the
  parameters and are called basic variables or
  leading variables.

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Example
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Example
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Example
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Solve the System
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Solve the System
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Echelon Form

• A rectangular matrix is in row reduced echelon
  form if it has the following properties
• 1. It is in echelon form.
• 2. All entries in a column above and below a
  leading entry are zero.
• 3. Each leading entry is a 1, the only nonzero
  entry in its column.

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Reduced Row Echelon Form
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Reduced Row Echelon Form
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Solve the System
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Solve the System
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Example
Estimate the temperatures T1, T2, T3, T4, T5, and
T6 at the six points on the steel plate below.
The value Tk is approximated by the average value
of the temperature at the four closest points.
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T1
T2
T3
0
0
T6
T5
T4
0
0
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Rank

• The rank of a matrix is the number of nonzero
  rows in its row echelon form.
• Rank Theorem
• Let A be the coefficient matrix of a system of
  linear equations with n variables. If the system
  is consistent, then
• Number of free variable n rank(A)

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Homogeneous System
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Theorem