Essential Vocabulary
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Augmented matrix
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.
