Linear Equations and Reduced Row Echelon Form
Solving system of linear equation
equivalent
A
x
=
b
A’=
[
A
b
]
A
complex
system of
linear equations
……
A’’
A’’’
R=
[
R
b
]
R
x
=
b
A
simple
system of
linear equations
1. Interchange any two rows of the matrix
2. Multiply every entry of some row by the same nonzero scalar
3. Add a multiple of one row of the matrix to another row
Reduced Row
Echelon Form (RREF)
elementary row operations:
Reduced Row Echelon Form
•
A system of linear equations is easily solvable if its
augmented matrix is in
reduced row echelon form
•
Row Echelon Form (REF)
1. Each nonzero row lies
above
every zero row
2.
The
leading entries
are
in echelon form
階層
Reduced Row Echelon Form
•
A system of linear equations is easily solvable if its
augmented matrix is in
reduced row echelon form
•
Row Echelon Form (REF)
1. Each nonzero row lies
above
every zero row
No zero rows
2.
The
leading entries
are
in echelon form
NO
Reduced Row Echelon Form
•
A system of linear equations is easily solvable if its
augmented matrix is in
reduced row echelon form
•
Reduced Row Echelon Form (RREF)
1-2 The matrix is in row
echelon form
3. The columns containing
the
leading entries
are
standard vectors
.
Reduced Row Echelon Form
•
A system of linear equations is easily solvable if its
augmented matrix is in
reduced row echelon form
•
Reduced Row Echelon Form
(RREF)
1-2 The matrix is in row
echelon form
3. The columns containing
the
leading entries
are
standard vectors
.
Reduced Row Echelon Form
A
R
Leading Entry
The
pivot positions
of A are
(1,1)
,
(2,3)
and
(3,4)
.
The
pivot columns
of A are
1
st
,
3
rd
and
4
th
columns.
Pivot
中樞
RREF is unique!
•
A matrix can be transformed into multiple REF by row
operation, but only one RREF
REF
RREF
REF
REF
Not going to proof
Solving systems of linear equations using Reduced Row Echelon Form (RREF) is a powerful technique in linear algebra. By transforming matrices into RREF, we can easily solve both simple and complex systems of equations. The process involves elementary row operations like row interchange, scalar multiplication, and row addition. The key to efficient solving lies in ensuring that the augmented matrix is in reduced row echelon form, where nonzero rows are above zero rows and leading entries are in echelon form.
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Presentation Transcript
Reduced Row Echelon Form (RREF)
Solving system of linear equation A simple system of linear equations A complex system of linear equations R x = b Ax = b equivalent R=[ R b ] A =[ Ab ] A A Reduced Row Echelon Form (RREF) elementary row operations: 1. Interchange any two rows of the matrix 2. Multiply every entry of some row by the same nonzero scalar 3. Add a multiple of one row of the matrix to another row
Reduced Row Echelon Form A system of linear equations is easily solvable if its augmented matrix is in reduced row echelon form Row Echelon Form (REF) 1. Each nonzero row lies above every zero row 2. The leading entries are in echelon form
Reduced Row Echelon Form A system of linear equations is easily solvable if its augmented matrix is in reduced row echelon form Row Echelon Form (REF) NO 1. Each nonzero row lies above every zero row 2. The leading entries are in echelon form No zero rows
Reduced Row Echelon Form A system of linear equations is easily solvable if its augmented matrix is in reduced row echelon form Reduced Row Echelon Form (RREF) 1-2 The matrix is in row echelon form 3. The columns containing the leading entries are standard vectors.
Reduced Row Echelon Form A system of linear equations is easily solvable if its augmented matrix is in reduced row echelon form Reduced Row Echelon Form (RREF) 1-2 The matrix is in row echelon form 3. The columns containing the leading entries are standard vectors.
Pivot Reduced Row Echelon Form A R Leading Entry The pivot positions of A are (1,1), (2,3) and (3,4). The pivot columns of A are 1st, 3rd and 4th columns.
Not going to proof RREF is unique! A matrix can be transformed into multiple REF by row operation, but only one RREF REF 2 3 0 1 0 0 1 3 1 3 6 3 RREF REF 1 0 0 2 3 0 1 3 3 3 6 9 REF 1 0 0 2 3 0 1 0 3 3 15 9
