Essential Vocabulary

Here is a link to a list of these terms on Quizlet

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Augmented matrix

Every linear system
\[\begin{array}{ccccccccc} a_{11}x_1 &+ &a_{12}x_2&+&\ldots& +&a_{1n}x_n&= &b_1 \\ a_{21}x_1 &+ &a_{22}x_2&+&\ldots& +&a_{2n}x_n&= &b_2 \\ &&&&\vdots&&&& \\ a_{m1}x_1 &+ &a_{m2}x_2&+&\ldots& +&a_{mn}x_n&= &b_m \end{array}\]
can be written in the augmented matrix form as follows:
\[\left [\begin{array}{cccc|c} a_{11}&a_{12}&\ldots&a _{1n}&b_1\\a_{21}&a_{22}&\ldots&a _{2n}&b_2\\\vdots& \vdots& \ddots& \vdots& \vdots \\a_{m1}&a_{m2}&\ldots&a _{mn}&b_m \end{array}\right ]\]
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 \(A\). The vertical array to the right of the bar is called a constant vector.
\[A=\begin{bmatrix}a_{11}&a_{12}&\ldots&a _{1n}\\a_{21}&a_{22}&\ldots&a _{2n}\\\vdots& \vdots& \ddots& \vdots \\a_{m1}&a_{m2}&\ldots&a _{mn}\end{bmatrix}\quad \text {and}\quad \vec {b}=\begin{bmatrix}b_1\\b_2\\\vdots \\b_m\end{bmatrix}\]
We will sometimes use the following notation to represent an augmented matrix.
\[\left [\begin{array}{c|c} A & \vec {b}\\ \end{array}\right ]\]

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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.

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Basic variable

See Leading variable

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Coefficient matrix

A coefficient matrix is a matrix whose entries are the coefficients of a system of linear equations. For the system
\[\begin{array}{ccccccccc} a_{11}x_1 &+ &a_{12}x_2&+&\ldots& +&a_{1n}x_n&= &b_1 \\ a_{21}x_1 &+ &a_{22}x_2&+&\ldots& +&a_{2n}x_n&= &b_2 \\ &&&&\vdots&&&& \\ a_{m1}x_1 &+ &a_{m2}x_2&+&\ldots& +&a_{mn}x_n&= &b_m \end{array},\]
the coefficient matrix is \(A=\begin{bmatrix}a_{11}&a_{12}&\ldots&a _{1n}\\a_{21}&a_{22}&\ldots&a _{2n}\\\vdots& \vdots& \ddots& \vdots \\a_{m1}&a_{m2}&\ldots&a _{mn}\end{bmatrix}\).

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Consistent system

A system of equations that has at least one solution.

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Convergence

when the iterates of an iterative method approach a solution

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Divergence

when the iterates of an iterative method fail to approach a solution

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Elementary row operations

The following three operations performed on a linear system are called elementary row operations.
1.
Switching the order of equations (rows) \(i\) and \(j\):
\[R_i\leftrightarrow R_j\]
2.
Multiplying both sides of equation (row) \(i\) by the same non-zero constant, \(k\), and replacing equation \(i\) with the result:
\[kR_i\rightarrow R_i\]
3.
Adding \(k\) times equation (row) \(i\) to equation (row) \(j\), and replacing equation \(j\) with the result:
\[R_j+kR_i\rightarrow R_j\]

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Equivalent linear systems

Linear systems are called equivalent if they have the same solution set.

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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.

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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.

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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.

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Gaussian elimination

The process of using the elementary row operations on a matrix to transform it into row-echelon form is called Gaussian Elimination.

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Inconsistent system

A system of equations that has no solution.

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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.

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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.

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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.

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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.

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Linear equation

A linear equation in variables \(x_1, \ldots , x_n\) is an equation that can be written in the form
\[a_1x_1+a_2x_2+\ldots +a_nx_n=b\]
where \(a_1,\ldots ,a_n\) and \(b\) are constants.

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Pivot

In Gaussian elimination, an entry chosen to become a leading coefficient used to get zeros in the remaining rows.

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Rank of a matrix

The rank of matrix \(A\), denoted by \(\mbox {rank}(A)\), is the number of nonzero rows that remain after we reduce \(A\) to row-echelon form by elementary row operations.

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Reduced row echelon form

A matrix that is already in row-echelon form is said to be in reduced row-echelon form if:
1.
Each leading entry is \(1\)
2.
All entries above and below each leading \(1\) are \(0\)

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Row echelon form

A matrix is said to be in row-echelon form if:
1.
All entries below each leading entry are 0.
2.
Each leading entry is in a column to the right of the leading entries in the rows above it.
3.
All rows of zeros, if there are any, are located below non-zero rows.

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Row equivalent matrices

Two matrices \(A\) and \(B\) are said to be row equivalent if there is a sequence of elementary row operations that converts \(A\) to \(B\).

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System of linear equations

A finite set of linear (degree 1) equations each with the same variables.

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