Systems of linear equations

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Systems of linear equations
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Simple system
Solution
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Not so simple system
Solve
Solution
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Linear equations
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Examples
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Example
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Linear systems
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Consistency
A system of equations that has no solution is
said to be inconsistent if there is at least one
solution of the system, it is said to be
consistent.
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Solution possibilities for two lines
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A deep statement
Every system of linear equations has either no
solutions, exactly one solution, or infinitely
many solutions.
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Augmented matrices
This is called the augmented matrix for the
system.
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Example
Do handout Q1-Q8
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Solving a system
The main idea here is to replace the given system
by a new system that has the same solution set
but which is easier to solve.
We apply the following three types of operations
to eliminate unknowns systematically 1.
Multiply an equation by a non-zero constant. 2.
Interchange two equations. 3. Add a multiple of
one equation to another.
These operations correspond to the following
elementary row operations on the augmented
matrix 1. Multiply a row by a non-zero
constant. 2. Interchange two rows. 3. Add a
multiple of one row to another row.
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An example
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An example
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An example
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Echelon form
We wish to reduce our matrices to those with the
following properties 1. If a row does not
consist entirely of zeros, then the first nonzero
number in the row is a 1. (This is called a
leading 1.) 2. If there are any rows that consist
entirely of zeros, then they are grouped together
at the bottom of the matrix. 3. In any two
successive rows that do not consist entirely of
zeros, the leading 1 in the lower rows occurs
farther to the right than the leading row in the
higher row. 4. Each column with a leading 1 has
zeros everywhere else.
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Echelon form
Echelon matrices 1. If a row does not consist
entirely of zeros, then the first nonzero
number in the row is a 1. (This is called a
leading 1.) 2. If there are any rows that consist
entirely of zeros, then they are grouped together
at the bottom of the matrix. 3. In any two
successive rows that do not consist entirely of
zeros, the leading 1 in the lower rows occurs
farther to the right than the leading row in the
higher row. 4. Each column with a leading 1 has
zeros everywhere else.
A matrix having properties 1, 2 and 3 (but not
necessarily 4) is said to be in row-echelon form.
A matrix having properties 1, 2, 3 and 4 is said
to be in reduced row-echelon form.
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Reduced versus non-reduced
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From echelon form to solution(Example 1 unique
solution)
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From echelon form to solution(Example 2
infinite solutions)
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From echelon form to solution(Example 3 no
solutions)
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Gaussian elimination an example
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Gaussian elimination an example
The matrix is now in row-echelon form!
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Gaussian elimination an example
The matrix is now in reduced row-echelon form!
The above procedure for reducing a matrix to
reduced row-echelon form is called Gauss-Jordan
elimination.
If we use only the first five steps, the
procedure produces a row-echelon form and is
called Gaussian elimination.
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Not so simple system
Augmented matrix
Solution
Row-echelon form
Reduced row-echelon form
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Back-substitution
Row-echelon form
Solving for the lead variables
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Homogeneous linear systems
So a homogenous system either has only the
trivial solution, or infinitely many solutions in
addition to the trivial solution. All such
systems are consistent.
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Homogeneous linear systems
Do Handout Q1-Q26
Theorem A homogeneous system of linear equations
with more unknowns than equations has infinitely
many solutions.
Note A consistent nonhomogeneous system with
more unknowns than equations also has infinitely
many solutions.
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