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Section 2.3 Important Definitions for Linear Equations

Subsection 2.3.1 The Definition of a Linear Equation

The equations given in the in Subsection 2.1.1 are examples of linear equations in two variables. A linear equation in \(n\) variables \(x_1, x_2, \dots, x_n\) is an equation of the form

\begin{equation*} a_1x_1+ a_2x_2+ \cdots +a_n x_n= b \end{equation*}

where \(a_1,\dots,a_n\) and \(b\) are constants. The numbers \(a_1,a_2,\dots,a_n\) are called the coefficients of the equation, that is, \(a_1\) is the coefficient of \(x_1\text{,}\) \(a_2\) is the coefficient of \(x_2\text{,}\) etc. This is also called a linear equation in \(n\) unknowns.

The following are examples of linear equations:

  • \(2x=4\) (one variable)

  • \(4x-2y=7\) (two variables)

  • \(\tfrac12 x -\tfrac35 y +\pi z=0\) (three variables)

  • \(x_1-3x_2+4x_3-7x_4=100\) (four variables: a coefficient of \(1\) is usually omitted)

  • \(3x_1+15x_5=-10\) (five variables: the ones with a coefficient of zero are not written)

Subsection 2.3.2 The Definition of a System of Linear Equations

A system of \(m\) linear equations in \(n\) unknowns is a list of \(m\) linear equations, each of which has the same set of \(n\) unknowns. They are usually presented with the purpose of finding all of their simultaneous solutions.

The original example given in Subsection 2.1.1 is a system of 2 equations in 2 unknowns. Here are some more examples:

  • A system of 2 equations in 3 unknowns:

    \begin{align*} x-y+z \amp = 1\\ 2x+3y-z \amp = 2 \end{align*}

  • A system of 3 equations in 2 unknowns:

    \begin{align*} u+v \amp = 1\\ 2u+v \amp = 3\\ u+2v \amp =3 \end{align*}

  • A system of 4 equations in 5 unknowns:

    \begin{alignat*}{5} x_1 \amp \ +\ \amp x_2 \amp \amp \amp \amp \amp \amp \amp \ =\ 1\\ \amp \amp x_2 \amp \ +\ \amp x_3 \amp \amp \amp \amp \amp \ =\ 2\\ \amp \amp \amp \amp x_3 \amp \ +\ \amp x_4 \amp \amp \amp \ =\ 3\\ \amp \amp \amp \amp \amp \amp x_4 \amp \ +\ \amp x_5 \amp \ =\ 4 \end{alignat*}

  • A system of \(m\) equations in \(n\) unknowns \(x_1, x_2, x_3\dots,x_n\text{:}\)

    \begin{gather*} a_{1,1}x_1 + a_{1,2}x_2 + \cdots + a_{1,n}x_n = b_1\\ a_{2,1}x_1 + a_{2,2}x_2 + \cdots + a_{2,n}x_n = b_2\\ \vdots\\ a_{m,1}x_1 + a_{m,2}x_2 + \cdots + a_{m,n}x_n = b_m \end{gather*}

Subsection 2.3.3 The Coefficient and Augmented Matrix of a System of Linear Equations

A matrix is a rectangular array of numbers (the plural is matrices). Here is a matrix:

\begin{equation*} \begin{bmatrix} 1 \amp 2 \amp 3 \amp 4\\ 5 \amp 6 \amp 7 \amp 8\\ 9 \amp 10\amp 11\amp 12 \end{bmatrix} \end{equation*}

This matrix has 3 rows:

  • \(\begin{bmatrix} 1 \amp 2 \amp 3 \amp 4 \end{bmatrix}\) is row 1,
  • \(\begin{bmatrix} 5 \amp 6 \amp 7 \amp 8 \end{bmatrix}\) is row 2 and
  • \(\begin{bmatrix} 9 \amp 10\amp 11\amp 12 \end{bmatrix}\) is row 3.

Similarly, the matrix has 4 columns:

  • \(\begin{bmatrix} 1\\ 5\\ 9 \end{bmatrix}\) is column \(1\)
  • \(\begin{bmatrix} 2\\ 6\\ 10 \end{bmatrix}\) is column \(2\)
  • \(\begin{bmatrix} 3 \\ 7\\ 11 \end{bmatrix}\) is column \(3\text{,}\) and
  • \(\begin{bmatrix} 4\\ 8\\ 12 \end{bmatrix}\) is column \(4\)

We call a matrix with 3 rows and 4 columns a \(3\times4\) matrix.

More generally, a matrix with \(m\) rows and \(n\) columns is called an \(m\times n\) matrix. An \(m\times n\) matrix \(A\) has the form

\begin{equation*} A=\begin{bmatrix} a_{1,1} \amp a_{1,2} \amp \cdots \amp a_{1,n}\\ a_{2,1} \amp a_{2,2} \amp \cdots \amp a_{2,n}\\ \amp \amp \vdots\\ a_{m,1} \amp a_{m,2} \amp \cdots \amp a_{m,n} \end{bmatrix} \end{equation*}

This notation means that \(a_{i,j}\) is the number in both row \(i\) and column \(j\text{.}\) We will call the rows \(R_1, R_2,\ldots,R_m\) and the columns \(C_1, C_2, \ldots,C_n\text{.}\) In other words,

\begin{equation*} R_i =\begin{bmatrix} a_{i,1} \amp a_{i,2} \amp a_{i,3}, \amp \cdots \amp a_{i,n}\end{bmatrix}, \end{equation*}

and

\begin{equation*} C_j=\begin{bmatrix} a_{1,j}\\ a_{2,j}\\ a_{3,j}\\ \vdots\\ a_{m,j} \end{bmatrix} \end{equation*}

The notation for this matrix is similar to that used for a system of linear equations, and for good reason. Suppose we have a system of \(m\) linear equations in \(n\) unknowns:

\begin{equation*} \begin{array}{lcl} a_{1,1}x_1 + a_{1,2}x_2 +{} \amp \cdots \amp {}+ a_{1,n}x_n = b_1\\ a_{2,1}x_1 + a_{2,2}x_2 +{} \amp \cdots \amp {}+ a_{2,n}x_n = b_2\\ \amp \vdots\\ a_{m,1}x_1 + a_{m,2}x_2 +{} \amp \cdots \amp {}+ a_{m,n}x_n = b_m \end{array} \end{equation*}

The coefficient matrix \(A\) is then the \(m\times n\) matrix

\begin{equation*} A=\begin{bmatrix} a_{1,1} \amp a_{1,2} \amp \cdots \amp a_{1,n}\\ a_{2,1} \amp a_{2,2} \amp \cdots \amp a_{2,n}\\ \amp \amp \vdots\\ a_{m,1} \amp a_{m,2} \amp \cdots \amp a_{m,n} \end{bmatrix} \end{equation*}

and the augmented matrix of the system is the \(m\times( n+1)\) matrix

\begin{equation*} A=\begin{bmatrix} a_{1,1} \amp a_{1,2} \amp \cdots \amp a_{1,n}\amp b_1\\ a_{2,1} \amp a_{2,2} \amp \cdots \amp a_{2,n}\amp b_2\\ \amp \amp \vdots\amp \amp \\ a_{m,1} \amp a_{m,2} \amp \cdots \amp a_{m,n}\amp b_m \end{bmatrix} \end{equation*}

Hence the augmented is the coefficient matrix with one column (the constants on the right side of the equations) added. To emphasize the extra column, the augmented matrix is sometimes written as

\begin{equation*} A=\left[\begin{array}{llcl|l} a_{1,1} \amp a_{1,2} \amp \cdots \amp a_{1,n}\amp b_1\\ a_{2,1} \amp a_{2,2} \amp \cdots \amp a_{2,n}\amp b_2\\ \amp \amp \vdots\amp \amp \\ a_{m,1} \amp a_{m,2} \amp \cdots \amp a_{m,n}\amp b_m \end{array}\right] \end{equation*}

Here are some examples of coefficient matrices and augmented matrices:

  • A system of 2 equations in 3 unknowns:

    \begin{align*} x-y+z \amp = 1\\ 2x+3y-z \amp = 2 \end{align*}

    Coefficient matrix:

    \begin{equation*} \begin{bmatrix} 1\amp -1\amp 1\\ 2\amp 3\amp -1 \end{bmatrix} \end{equation*}

    Augmented matrix:

    \begin{equation*} \begin{bmatrix} 1\amp -1\amp 1\amp 1\\ 2\amp 3\amp -1\amp 2 \end{bmatrix} \end{equation*}

  • A system of 3 equations in 2 unknowns:

    \begin{align*} u+v \amp = 1\\ 2u+v \amp = 3\\ u+2v \amp =3 \end{align*}

    Coefficient matrix: \(\begin{bmatrix} 1\amp 1\\ 2\amp 1\\ 1\amp 2 \end{bmatrix}\)

    Augmented matrix: \(\left[\begin{array}{cc|c} 1\amp 1\amp 1\\ 2\amp 1\amp 3\\ 1\amp 2\amp 3 \end{array}\right]\)

  • A system of 4 equations in 5 unknowns

    \begin{alignat*}{5} x_1 \amp \ +\ \amp x_2 \amp \amp \amp \amp \amp \amp \amp \ =\ 1\\ \amp \amp x_2 \amp \ +\ \amp x_3 \amp \amp \amp \amp \amp \ =\ 2\\ \amp \amp \amp \amp x_3 \amp \ +\ \amp x_4 \amp \amp \amp \ =\ 3\\ \amp \amp \amp \amp \amp \amp x_4 \amp \ +\ \amp x_5 \amp \ =\ 4 \end{alignat*}

    Coefficient matrix: \(\begin{bmatrix} \begin{array}{rrrrr} 1\amp 1\amp 0\amp 0\amp 0\\ 0\amp 1\amp 1\amp 0\amp 0\\ 0\amp 0\amp 1\amp 1\amp 0\\ 0\amp 0\amp 0\amp 1\amp 1 \end{array} \end{bmatrix}\)

    Augmented matrix: \(\left[\begin{array}{ccccc|c} 1\amp 1\amp 0\amp 0\amp 0\amp 1\\ 0\amp 1\amp 1\amp 0\amp 0\amp 2\\ 0\amp 0\amp 1\amp 1\amp 0\amp 3\\ 0\amp 0\amp 0\amp 1\amp 1\amp 4 \end{array}\right]\)

Exercises Exercises

1.

Which of the following equations are linear?

  1. \(\displaystyle x+y-z^2=0\)

  2. \(\displaystyle 2x+yz=1\)

  3. \(\displaystyle z=3\)

  4. \(\displaystyle \pi^2 x-\sqrt[3]{3}y=8\sqrt{5}\)

  5. \(\displaystyle 3x-4y=\frac73 z-5+t\)

  6. \(\displaystyle \cos{x}=3y+2z\)

Solution

  1. Not linear because of \(z^2\)

  2. Not linear because of \(yz\)

  3. Linear

  4. Linear (the exponents involve the constants, but not the variables)

  5. Linear (Tricky because variables are on both sides of the equal sign. It is the same as \(3x-4y-\frac73 z-t =-5 \text{.}\))

  6. Not linear because of the \(\cos\) function

2.

Consider the following system of linear equations:

\begin{equation*} 2x-4y+z=-1\\ 3x-3y+z=1 \end{equation*}

Which of the following are solutions to this system

  • \(\displaystyle (x,y,z)=(2,1,-1) \)

  • \(\displaystyle (x,y,z)=(1,1,1) \)

  • \(\displaystyle (x,y,z)=(0,0,1) \)

Solution

  • Substitute \((x,y,z)=(2,1,-1) \) into the equations:

    \begin{equation*} 2x-4y+z=4-4-1=-1\\ 3x-3y+z=6-3-1=2 \not=1 \end{equation*}

    and so \((2,1,-1)\) is not a solution.

  • Substitute \((x,y,z)=(1,1,1) \) into the equations:

    \begin{equation*} 2x-4y+z=2-4+1=-1\\ 3x-3y+z=3-3+1=1 \end{equation*}

    and so \((1,1,1)\) is a solution.

  • Substitute \((x,y,z)=(0,0,1) \) into the equations:

    \begin{equation*} 2x-4y+z=0+0+1\not=-1\\ 3x-3y+z=0+0+1=1 \end{equation*}

    and so \((0,0,1)\) is not a solution.

3.

Give a system of two equations in two unknowns that has \((1,2)\) as its unique solution.

Solution

There are many solutions corresponding to two lines intersecting at \((1,2)\text{.}\) Surely the easiest is

\begin{align*} x\amp=1\\ y\amp=2 \end{align*}
4.

Give the coefficient matrix and the augmented matrix this system of linear equations:

\begin{align*} x+y \amp= 2 \\ x-y \amp= 0 \end{align*}
Solution

\(\begin{bmatrix} 1\amp1\\1\amp-1 \end{bmatrix}\) and \(\left[\begin{array}{cc|c} 1 \amp 1 \amp 2 \\ 1 \amp -1 \amp 0 \end{array}\right]\)

5.

Give the coefficient matrix and the augmented matrix this system of linear equations:

\begin{align*} x_1+x_2-x_3+x_5\amp= 4 \\ 2x_1-3x_2 +x_4\amp= 7 \end{align*}
Solution

\(\begin{bmatrix} 1 \amp 1 \amp -1 \amp 0 \amp 1 \\ 2\amp-3 \amp 0 \amp 1 \amp 0 \end{bmatrix}\) and \(\left[\begin{array}{ccccc|c} 1 \amp 1 \amp -1 \amp 0 \amp 1 \amp 4\\ 2\amp-3 \amp 0 \amp 1 \amp 0 \amp7 \end{array}\right]\)

6.

Give the coefficient matrix and the augmented matrix this system of linear equations:

\begin{align*} x+y-z \amp=1\\ 3x+y\amp= 2\\ y+z \amp=3\\ x+z\amp=4\\ x+y+z\amp=5 \end{align*}
Solution

\(\begin{bmatrix} 1 \amp 1 \amp -1 \\ 3 \amp 1\amp 0 \\ 0 \amp 1 \amp 1\\ 1 \amp 0 \amp 1 \\ 1 \amp 1 \amp 1 \end{bmatrix}\) and \(\left[\begin{array}{ccc|c} 1 \amp 1 \amp -1 \amp 1\\ 3 \amp 1\amp 0 \amp 2\\ 0 \amp 1 \amp 1\amp 3\\ 1 \amp 0 \amp 1 \amp 4\\ 1 \amp 1 \amp 1\amp 5 \end{array}\right]\)

7.

Give the system of equations whose augmented matrix is

\begin{equation*} \left[\begin{array}{cc|c} 1 \amp 4 \amp 1\\ 3 \amp -2\amp 0 \\ 0 \amp 1 \amp 1\\ -1 \amp 1 \amp 3 \end{array}\right] \end{equation*}
Solution
\begin{align*} x+4y\amp= 1\\ 3x-2y\amp=0\\ y\amp=1\\ -x+y\amp=3 \end{align*}
8.

There is a special matrix \(I_n\) called the identity matrix. It has \(n\) rows and \(n\) columns. The entries \(a_{i,j}\) of the matrix satisfy

\begin{equation*} a_{i,j}= \begin{cases} 1 \amp \text{if } i=j\\ 0 \amp \text{if } i\not=j\\ \end{cases} \end{equation*}

  1. Show the the matrix \(I_n\) has the value \(1\) along the (main) diagonal going from the upper left corner to the lower right corner of the matrix, and has the value \(0\) elsewhere.

  2. Suppose that a system of linear equations has an augmented matrix of the form

    \begin{equation*} \left[ I_n \mid B \right] \end{equation*}

    where

    \begin{equation*} B= \begin{bmatrix} b_1\\b_2\\ \vdots\\b_n \end{bmatrix} \end{equation*}

    What does the say about the number of variables (unknowns) for the corresponding system of linear equations, and what does it say about the solutions to the system?

Solution

  1. When \(i=j\text{,}\) the row number and column number are identical, and so the entry \(a_{i.j}\) is on the diagonal. This implies that

    \begin{equation*} a_{i,j}= \begin{cases} 1 \amp \text{ for entries on the main diagonal}\\ 0 \amp \text{ for entries not on the main diagonal} \\ \end{cases} \end{equation*}

  2. The number of unknowns is \(n\text{,}\) the number of columns in the matrix. We may call them \(x_1, x_2,\dots,x_n\text{.}\) The augmented matrix now says that

    \begin{align*} x_1\amp =b_1\\ x_2\amp =b_2\\ \amp \vdots\\ x_n\amp =b_n \end{align*}