Glossary of Linear Algebra Terms

Branko Ćurgus

linear equation

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ and $b$ be real numbers. A linear equation with $n$ unknowns $x_1, x_2, \ldots, x_n$ is an equation of the form \[ a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b. \] The given real numbers $a_1, a_2, \ldots, a_n$ and $b$ are called the coefficients of the linear equation.
Comments
  1. It is crucial to make a conceptual distinction between coefficients and unknowns; the coefficients are given ("known") numbers, while unknowns are symbols whose numerical values are to be determined. To emphasize this conceptual distinction, sometimes I like to write equations in color: \[ \color{green}{a_1} \color{red}{x_1} + \color{green}{a_{2}} \color{red}{x_{2}} + \cdots + \color{green}{a_{n}} \color{red}{x_{n}} = \color{green}{b}. \] The coefficients are in friendly green and unknowns are in scary red.
  2. Three specific examples of linear equations with one unknown are \[ 2 \color{red}{x_1} = 3, \quad 0 \color{red}{x_1} = 0, \quad 0 \color{red}{x_1} = 1. \]
  3. Three specific examples of linear equations with two unknowns are \[ 2 \color{red}{x_1} +3 \color{red}{x_2} = 7, \quad 0 \color{red}{x_1} + \color{red}{x_2} = 1, \quad 0 \color{red}{x_1} + 0 \color{red}{x_2} = 1. \]

system of linear equations

Let $m$ and $n$ be a positive integers. A collection of $m$ linear equations each with $n$ unknowns \begin{alignat*}{4} \color{green}{a_{11}} \color{red}{x_1} &+& \color{green}{a_{12}} \color{red}{x_{2}} &+& \ \cdots \ &+& \color{green}{a_{1n}} \color{red}{x_{n}} & = \color{green}{b_{1}} \\ \color{green}{a_{21}} \color{red}{x_1} &+& \color{green}{a_{22}} \color{red}{x_{2}} &+& \ \cdots \ &+& \color{green}{a_{2n}} \color{red}{x_{n}} & = \color{green}{b_{2}} \\ & & & & & & & \ \ \vdots \\ \color{green}{a_{m1}} \color{red}{x_1} &+& \color{green}{a_{m2}} \color{red}{x_{2}} &+& \ \cdots \ &+& \color{green}{a_{mn}} \color{red}{x_{n}} & = \color{green}{b_{m}} \\ \end{alignat*} is called a system of $m$ linear equations with $n$ unknowns. The real numbers colored green, that is the numbers $\color{green}{a_{11}},$ $\color{green}{a_{12}}, \ldots, \color{green}{a_{1n}},$ $\color{green}{b_{1}},$ $\color{green}{a_{21}},$ $\color{green}{a_{22}},\ldots, \color{green}{a_{2n}},$ $\color{green}{b_{2}}, \ldots, \color{green}{a_{m1}},$ $\color{green}{a_{m2}} \ldots \color{green}{a_{mn}},$ $\color{green}{b_{m}}$ are called the coefficients of the system of linear equations. The red symbols $\color{red}{x_1},$ $\color{red}{x_{2}}, \ldots, \color{red}{x_{n}}$ are called the unknowns of the system of linear equations.
Comments

vector equation

Let $m$ and $n$ be a positive integers. Let $\color{green}{\mathbf{a}_1}, \ldots, \color{green}{\mathbf{a}_n}$ and $\color{green}{\mathbf{b}}$ be $n+1$ given vectors in $\mathbb{R}^m$. An algebraic expression of the form \begin{equation*} \color{green}{\mathbf{a}_1} \color{red}{x_1} + \cdots + \color{green}{\mathbf{a}_n} \color{red}{x_n} = \color{green}{\mathbf{b}} \end{equation*} is called a vector equation with $n$ unknowns. The vectors $\color{green}{\mathbf{a}_1}, \ldots, \color{green}{\mathbf{a}_n}$ and $\color{green}{\mathbf{b}}$ are called coefficients of the vector equation. The red symbols $\color{red}{x_1}, \ldots, \color{red}{x_{n}}$ are called the unknowns of the vector equation.
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matrix equation

Let $m$ and $n$ be a positive integers.
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triumvirate of equations

Let $m$ and $n$ be a positive integers.
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zero matrix

A matrix whose all entries are zero is called a zero matrix.
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  1. The $2\times 3$ zero matrix is $\displaystyle \left[\! \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \!\right].$

zero row

A row whose all entries are zero is called a zero row.
Comments
  1. The second row of the following $4\times 3$ matrix $\displaystyle \left[\! \begin{array}{ccc} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{array} \!\right]$ is a zero row.

leading entry of a row, leading zeros of a row

The leftmost nonzero entry of a nonzero row is called the leading entry of a row. The zeros to the left of the leading entry are called the leading zeros of a row. All entries of a zero row are leading zeros.
Comments
  1. Consider the following $4\!\times\!3$ matrix is $\displaystyle \left[\! \begin{array}{ccc} 3 & 2 & 1 \\ 0 & 0 & 0 \\ 0 & 2 & 3 \\ 0 & 0 & 1 \end{array} \!\right].$ The leading entry of the first row is $3$, the second row does not have a leading entry, the leading entry of the third row is $2$, and the leading entry of the fourth row is $1.$ The first row has $0$ leading zeros, the second row has $3$ leading zeros, the third row has $1$ leading zero, and the fourth row has $2$ leading zeros.

reduced row echelon form (RREF) of a matrix

Each zero matrix is in Reduced Row Echelon Form. A nonzero matrix is in Reduced Row Echelon Form if it satisfies the following three conditions:
  • Each nonzero row has strictly more leading zeros then the row above it.
  • The leading entry of each nonzero row is equal to $1$.
  • The leading entry of each nonzero row is the only nonzero entry in its column.
Comments
  1. The above three conditions imply that if a matrix is in reduced row echelon form, then all its zero rows are at the bottom.
  2. The $4\!\times\!3$ matrix $\displaystyle \left[\! \begin{array}{ccc} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{array} \!\right]$ is not in RREF since the third row (which is a nonzero row) has $1$ leading zero and the row above it (the second row) has three leading zeros. Hence the first condition is not satisfied. Also, the number $1$ is the leading entry in the third row and this $1$ is not the only nonzero entry in its column (the second column). The second column has two nonzero entries $1$ and $2.$
  3. The $4\!\times\!3$ matrix $\displaystyle \left[\! \begin{array}{ccr} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0\\ 0 & 0 & 0 \\ \end{array} \!\right]$ is in RREF. We first check the nonzero rows: the second row has $1$ leading zero and the row above it (the first row) has $0$ leading zeros. There are $2$ leading entries in the first and the second row and both are $1.$ Each leading entry is the only nonzero entry in its column. That is the first and the second column has the leading entries as the only nonzero entry.