MAT 2401 Linear Algebra

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MAT 2401Linear Algebra

• 1.1, 1.2 Part I Gauss-Jordan Elimination

http//myhome.spu.edu/lauw
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HW

• Written Homework

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Time

• Part I may be a bit longer today.
• Part II will be shorter next time.

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Preview

• Introduce the Matrix notations.
• Study the Elementary Row Operations.
• Study the Gauss-Jordan Elimination.

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Example 1
Elimination
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Example 1
Elimination Geometric Meaning
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How many solutions?

• Q Given a system of 2 equations in 2 unknowns,
  how many solutions are possible?

A
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How many solutions?

• Q Given a system of 3 equations in 3 unknowns,
  how many solutions are possible?

A
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How many solutions?

• Q Given a system of 3 equations in 3 unknowns,
  how many solutions are possible?

______ System ______ System
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Unique Solution

• We will focus only on systems of unique solution
  in part I.
• Such systems appear a lot in applications.

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Example 2
Elimination
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Observation 1
Q Why eliminations are not good? A 1. 2. 3.
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Observation 2
Compare the 2 systems
Q Are the 2 systems the same? A
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Observation 2
Compare the 2 systems
Q What do the 2 systems have in common? A
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Observation 2
Compare the 2 systems
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Observation 2
Compare the 2 systems
Q Which system is easier to solve? A
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Extreme Makeover?

• We want a solution method that
• it is systematic, extendable, and easy to
  automate
• it can transform a complicated system into a
  simple system

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Extreme Makeover?

• We want a solution method that
• it is systematic, extendable, and easy to
  automate
• it can transform a complicated system into a
  simple system

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Extreme Makeover?

• We want a solution method that
• it is systematic, extendable, and easy to
  automate
• it can transform a complicated system into a
  simple system

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Extreme Makeover?

• We want a solution method that
• it is systematic, extendable, and easy to
  automate
• it can transform a complicated system into a
  simple system

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Gauss-Jordan Elimination
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Gauss-Jordan Elimination

• Before we can describe our systematic solution
  method, we need the matrix notations.

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Essential Information

• A system can be represented compactly by a
  table of numbers.

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Matrix

• A matrix is a rectangular array of numbers.
• If a matrix has m rows and n columns, then the
  size of the matrix is said to be mxn.

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Example 2

• Write down the (Augmented) matrix representation
  of the given system.

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Coefficient Matrix

• The left side of the Augmented matrix is called
  the Coefficient Matrix.

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Elementary Row Operations

• We can perform the following operations on the
  matrix
• 1. Switching 2 rows.
• 2. Multiplying a row by a constant.
• 3. Adding a multiple of one row to another.

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Elementary Row Operations

• We can perform the following operations on the
  matrix
• 1. Switching 2 rows.

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Elementary Row Operations

• We can perform the following operations on the
  matrix
• 2. Multiplying a row by a constant.

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Elementary Row Operations

• We can perform the following operations on the
  matrix
• 3. Adding a multiple of one row to another.

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Elementary Row Operations

• Theory We can use the operations to simplify the
  system without changing the solution.
• 1. Switching 2 rows.
• 2. Multiplying a row by a constant.
• 3. Adding a multiple of one row to another.

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Elementary Row Operations

• Notations (examples)
• 1. Switching 2 rows.
• 2. Multiplying a row by a constant.
• 3. Adding a multiple of one row to another.

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Gauss-Jordan Elimination
Main Idea We want to use elementary row operations to get the matrix into the form (reduced row-echelon form RREF)
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Gauss-Jordan Elimination
Main Idea We want to use elementary row operations to get the matrix into the form (reduced row-echelon form RREF) The order of creating 0 and 1 is extremely important!
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Example 2
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Remarks

• Notice sometimes 2 parallel row operations can
  be done in the same step.
• The procedure (algorithm) is designed so that the
  exact order of creating the 0s and 1s is
  important.

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Remarks

• Try to avoid fractions!!

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How do I Confirm My Answers?
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Example 3

• Use Gauss-Jordan Elimination to solve the system.

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Example 3