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Every linear system
can be written in the augmented matrix form as follows:
The array to the left of the vertical bar is called the coefficient matrix of the linear
system and is often given a capital letter name, like . The vertical array to the right
of the bar is called a constant vector.
We will sometimes use the following notation to represent an augmented
matrix.
Back substitution
When a matrix is in row-echelon form, we can compute the solution to the system by
starting from the last equation and working backwards. This process is known as back
substitution.
Basic variable
See Leading variable
Coefficient matrix
A coefficient matrix is a matrix whose entries are the coefficients of a system of linear
equations. For the system
the coefficient matrix is .
Consistent system
A system of equations that has at least one solution.
Convergence
when the iterates of an iterative method approach a solution
Divergence
when the iterates of an iterative method fail to approach a solution
Elementary row operations
The following three operations performed on a linear system are called elementary
row operations.
(a)
Switching the order of equations (rows) and :
(b)
Multiplying both sides of equation (row) by the same non-zero constant,
, and replacing equation with the result:
(c)
Adding times equation (row) to equation (row) , and replacing equation
with the result:
Equivalent linear systems
Linear systems are called equivalent if they have the same solution set.
Free variable
When a linear system is in row-echelon form, the variables corresponding to columns
that do not have any leading coefficients (if there are any) are known as free
variables.
Gauss-Jordan elimination
The process of using the elementary row operations on a matrix to transform it into
reduced row-echelon form is called Gauss-Jordan elimination.
Gauss-Seidel Method
An iterative method for solving linear systems that is a refinement of the Jacobi
method, where we use computed values of variables alternately for quicker
convergence.
Gaussian elimination
The process of using the elementary row operations on a matrix to transform it into
row-echelon form is called Gaussian Elimination.
Inconsistent system
A system of equations that has no solution.
Iterative method
A technique where we repeat the same procedure (called an iteration) many times
(usually using a computer), and we obtain approximate solutions which we hope
“converge to” the actual solution.
Jacobi’s method
An iterative method for solving a system of equations where one variable is isolated
in each equation in order to compute the coordinate of the next iterate.
Leading entry (leading 1)
The first non-zero entry in a row of a matrix (when read from left to right) is called
the leading entry. When the leading entry is 1, we refer to it as a leading 1.
Leading variable (or Basic variable)
When a coefficient matrix is in row echelon form, a leading variable is a variable
corresponding to a column of the matrix with at least one leading entry.
Linear equation
A linear equation in variables is an equation that can be written in the
form
where and are constants.
Pivot
In Gaussian elimination, an entry chosen to become a leading coefficient used to get
zeros in the remaining rows.
Rank of a matrix
The rank of matrix , denoted by , is the number of nonzero rows that remain after we
reduce to row-echelon form by elementary row operations.
Reduced row echelon form
A matrix that is already in row-echelon form is said to be in reduced row-echelon
form if:
(a)
Each leading entry is
(b)
All entries above and below each leading are
Row echelon form
A matrix is said to be in row-echelon form if:
(a)
All entries below each leading entry are 0.
(b)
Each leading entry is in a column to the right of the leading entries in the
rows above it.
(c)
All rows of zeros, if there are any, are located below non-zero rows.
Row equivalent matrices
Two matrices and are said to be row equivalent if there is a sequence of elementary
row operations that converts to .
System of linear equations
A finite set of linear (degree 1) equations each with the same variables.
2024-09-06 02:12:21
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