Chapter%201:%20Linear%20Equations%20in%20Linear%20Algebra

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Chapter 1 Linear Equations in Linear Algebra

• 1.1 Systems of Linear Equations

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• Linear Equation
• System of linear equations A collection of one
  or more linear equations involving the same
  variables

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Exercise Determine if each of the following is
a system of linear equations.
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Definitions

• Solution Set The set of all possible solutions
  to a linear system.

• Two linear systems are called equivalent,
• if they have the same solution set.

Exercise Are the following systems equivalent?
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Matrix Notation

• Any linear system can be written as a matrix
  which is a rectangular array with m rows and n
  columns.
• The size of a matrix is the number of rows and
  columns it possesses. We write the size as mxn.

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• For the linear system
• The matrix is called the
• coefficient matrix and the matrix
• is called the augmented matrix.

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Algorithmic Procedure for Solving a Linear System

• Basic idea Replace one system with an
  equivalent system that is easier to solve.
• Example Solve the following system

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/ 2
/(-2)
R1(1/2) R1 R2(-1/2) R2
R1 R2
(-2)
)
-2R1R2 R2
/3
R2(1/3) R2
2R2R1 R1
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Elementary Row Operations

• (Replacement) Replace one row by the sum of
  itself and a multiple of another row.
• (Interchange) Interchange two rows.
• (Scaling) Multiply all entries in a row by a
  nonzero constant.

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Gauss-Jordan Elimination method to solve a
system of linear equations
Elementay Row Operations
Augmented matrix
Reduced augmented matrix
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Practice Elementary Row Operations
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Solutions to Linear Systems -- Existence and
Uniqueness

• Is the system consistent that is, does at least
  one solution exist?
• If a solution exists, is it the only one that
  is, is the solution unique?

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Example 1

• A graph of the following system of equations is
  shown to the right
• Is the system consistent?
• Is it unique?

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Example 2

• A graph of the following system of equations is
  shown to the right
• Is the system consistent?
• Is it unique?

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Example 3

• A graph of the following system of equations is
  shown to the right
• Is the system consistent?
• Is it unique?

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Example 4

• A graph of the following system of equations is
  shown to the right
• Is the system consistent?
• Is it unique?

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Find the solution set of each of the following
No solutions
Inconsistent
(2,1)
Unique
Consistent
Infinitely many solutions.
All (x,y) where
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Every system of linear equations has either

• No solutions, or
• Exactly one solution, or
• Infinitely many solutions.

Geometrically, what does this say about the
solution of a linear system?