TBIL-LA Linear Systems, Vector Equations, and Augmented Matrices (LE1)

🔗

Section 1.1 Linear Systems, Vector Equations, and Augmented Matrices (LE1)

🔗

Learning Outcomes

Translate back and forth between a system of linear equations, a vector equation, and the corresponding augmented matrix.

🔗

Subsection 1.1.1 Warm Up

🔗

Activity 1.1.1.

🔗

Consider the pairs of lines described by the equations below. Decide which of these are parallel, identical, or transverse (i.e., intersect in a single point).

🔗

(a)

🔗

\begin{aligned} - x_{1} + 3 x_{2} & = 1 \\ 2 x_{1} - 5 x_{2} & = 2 \end{aligned}

🔗

(b)

🔗

\begin{aligned} - x_{1} + 3 x_{2} & = 1 \\ 2 x_{1} - 6 x_{2} & = - 2 \end{aligned}

🔗

(c)

🔗

\begin{aligned} - x_{1} + 3 x_{2} & = 1 \\ 2 x_{1} - 6 x_{2} & = 3 \end{aligned}

🔗

Subsection 1.1.2 Class Activities

🔗

Definition 1.1.2.

🔗

A matrix is an

m \times n

array of real numbers with

m

rows and

n

columns:

[\begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{array}] = [\begin{array}{cccc} {\vec{v}}_{1} & {\vec{v}}_{2} & \dots & {\vec{v}}_{n} \end{array}] .

🔗

Frequently we will use matrices to describe an ordered list of its column vectors:

[\begin{matrix} a_{11} \\ a_{21} \\ ⋮ \\ a_{m 1} \end{matrix}], [\begin{matrix} a_{12} \\ a_{22} \\ ⋮ \\ a_{m 2} \end{matrix}], \dots, [\begin{matrix} a_{1 n} \\ a_{2 n} \\ ⋮ \\ a_{m n} \end{matrix}] = {\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{n} .

🔗

When order is irrelevant, we will use set notation:

{[\begin{matrix} a_{11} \\ a_{21} \\ ⋮ \\ a_{m 1} \end{matrix}], [\begin{matrix} a_{12} \\ a_{22} \\ ⋮ \\ a_{m 2} \end{matrix}], \dots, [\begin{matrix} a_{1 n} \\ a_{2 n} \\ ⋮ \\ a_{m n} \end{matrix}]} = {{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{n}} .

🔗

Definition 1.1.3.

🔗

A Euclidean vector is an ordered list of real numbers

[\begin{matrix} a_{1} \\ a_{2} \\ ⋮ \\ a_{n} \end{matrix}] .

🔗

We will find it useful to almost always typeset Euclidean vectors vertically, but the notation

{[\begin{array}{cccc} a_{1} & a_{2} & \dots & a_{n} \end{array}]}^{T}

is also valid when vertical typesetting is inconvenient. The set of all Euclidean vectors with

n

components is denoted as

R^{n},

and vectors are often described using the notation

\vec{v} .

🔗

Each number in the list is called a component, and we use the following definitions for the sum of two vectors, and the product of a real number and a vector:

[\begin{matrix} a_{1} \\ a_{2} \\ ⋮ \\ a_{n} \end{matrix}] + [\begin{matrix} b_{1} \\ b_{2} \\ ⋮ \\ b_{n} \end{matrix}] = [\begin{matrix} a_{1} + b_{1} \\ a_{2} + b_{2} \\ ⋮ \\ a_{n} + b_{n} \end{matrix}] c [\begin{matrix} a_{1} \\ a_{2} \\ ⋮ \\ a_{n} \end{matrix}] = [\begin{matrix} c a_{1} \\ c a_{2} \\ ⋮ \\ c a_{n} \end{matrix}]

🔗

Example 1.1.4.

🔗

Following are some examples of addition and scalar multiplication in

R^{4} .

[\begin{matrix} 3 \\ - 3 \\ 0 \\ 4 \end{matrix}] + [\begin{matrix} 0 \\ 2 \\ 7 \\ 1 \end{matrix}] = [\begin{matrix} 3 + 0 \\ - 3 + 2 \\ 0 + 7 \\ 4 + 1 \end{matrix}] = [\begin{matrix} 3 \\ - 1 \\ 7 \\ 5 \end{matrix}]

- 4 [\begin{matrix} 0 \\ 2 \\ - 2 \\ 3 \end{matrix}] = [\begin{matrix} - 4 (0) \\ - 4 (2) \\ - 4 (- 2) \\ - 4 (3) \end{matrix}] = [\begin{matrix} 0 \\ - 8 \\ 8 \\ - 12 \end{matrix}]

🔗

Definition 1.1.5.

🔗

A linear equation is an equation of the variables

x_{i}

of the form

a_{1} x_{1} + a_{2} x_{2} + \dots + a_{n} x_{n} = b .

🔗

A solution for a linear equation is a Euclidean vector

[\begin{matrix} s_{1} \\ s_{2} \\ ⋮ \\ s_{n} \end{matrix}]

🔗

that satisfies

a_{1} s_{1} + a_{2} s_{2} + \dots + a_{n} s_{n} = b

🔗

(that is, a Euclidean vector whose components can be plugged into the equation).

🔗

Remark 1.1.6.

🔗

In previous classes you likely used the variables

x, y, z

in equations. However, since this course often deals with equations of four or more variables, we will often write our variables as

x_{i},

and assume

x = x_{1}, y = x_{2}, z = x_{3}, w = x_{4}

when convenient.

🔗

Definition 1.1.7.

🔗

A system of linear equations (or a linear system for short) is a collection of one or more linear equations.

\begin{aligned} a_{11} x_{1} & + & a_{12} x_{2} & + & \dots & + & a_{1 n} x_{n} & = & b_{1} \\ a_{21} x_{1} & + & a_{22} x_{2} & + & \dots & + & a_{2 n} x_{n} & = & b_{2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ a_{m 1} x_{1} & + & a_{m 2} x_{2} & + & \dots & + & a_{m n} x_{n} & = & b_{m} \end{aligned}

🔗

Its solution set is given by

{[\begin{matrix} s_{1} \\ s_{2} \\ ⋮ \\ s_{n} \end{matrix}] | [\begin{matrix} s_{1} \\ s_{2} \\ ⋮ \\ s_{n} \end{matrix}] is a solution to all equations in the system} .

🔗

Remark 1.1.8.

🔗

When variables in a large linear system are missing, we prefer to write the system in one of the following standard forms:

🔗

Original linear system:

\begin{aligned} x_{1} + 3 x_{3} & = & 3 \\ 3 x_{1} - 2 x_{2} + 4 x_{3} & = & 0 \\ - x_{2} + x_{3} & = & - 2 \end{aligned}

🔗

Verbose standard form:

\begin{aligned} 1 x_{1} & + & 0 x_{2} & + & 3 x_{3} & = & 3 \\ 3 x_{1} & - & 2 x_{2} & + & 4 x_{3} & = & 0 \\ 0 x_{1} & - & 1 x_{2} & + & 1 x_{3} & = & - 2 \end{aligned}

🔗

Concise standard form:

\begin{aligned} x_{1} & + & 3 x_{3} & = & 3 \\ 3 x_{1} & - & 2 x_{2} & + & 4 x_{3} & = & 0 \\ - & x_{2} & + & x_{3} & = & - 2 \end{aligned}

🔗

Remark 1.1.9.

🔗

It will often be convenient to think of a system of equations as a vector equation.

🔗

By applying vector operations and equating components, it is straightforward to see that the vector equation

x_{1} [\begin{matrix} 1 \\ 3 \\ 0 \end{matrix}] + x_{2} [\begin{matrix} 0 \\ - 2 \\ - 1 \end{matrix}] + x_{3} [\begin{matrix} 3 \\ 4 \\ 1 \end{matrix}] = [\begin{matrix} 3 \\ 0 \\ - 2 \end{matrix}]

🔗

is equivalent to the system of equations

\begin{aligned} x_{1} & + & 3 x_{3} & = & 3 \\ 3 x_{1} & - & 2 x_{2} & + & 4 x_{3} & = & 0 \\ - & x_{2} & + & x_{3} & = & - 2 \end{aligned}

🔗

Definition 1.1.10.

🔗

A linear system is consistent if its solution set is non-empty (that is, there exists a solution for the system). Otherwise it is inconsistent.

🔗

Fact 1.1.11.

🔗

All linear systems are one of the following:

Consistent with one solution: its solution set contains a single vector, e.g. ${[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]}$
Consistent with infinitely-many solutions: its solution set contains infinitely many vectors, e.g. ${[\begin{matrix} 1 \\ 2 - 3 a \\ a \end{matrix}] | a \in R}$
Inconsistent: its solution set is the empty set, denoted by either ${}$ or $\emptyset .$

🔗

Activity 1.1.12.

🔗

All inconsistent linear systems contain a logical contradiction. Find a contradiction in this system to show that its solution set is the empty set.

\begin{aligned} - x_{1} + 2 x_{2} & = 5 \\ 2 x_{1} - 4 x_{2} & = 6 \end{aligned}

🔗

Activity 1.1.13.

🔗

Consider the following consistent linear system.

\begin{aligned} - x_{1} + 2 x_{2} & = - 3 \\ 2 x_{1} - 4 x_{2} & = 6 \end{aligned}

🔗

(a)

🔗

Find several different solutions for this system:

[\begin{matrix} 1 \\ - 1 \end{matrix}] [\begin{matrix} ? \\ 2 \end{matrix}] [\begin{matrix} 0 \\ ? \end{matrix}] [\begin{matrix} ? \\ ? \end{matrix}] [\begin{matrix} ? \\ ? \end{matrix}]

🔗

(b)

🔗

Suppose we let

x_{2} = a

where

a

is an arbitrary real number. Which of these expressions for

x_{1}

in terms of

a

satisfies both equations of the linear system?

🔗

$x_{1} = - 3 a$
$x_{1} = 3$
$x_{1} = 2 a + 3$
$x_{1} = - 4 a + 6$

Answer.

x_{1} = 2 a + 3

🔗

(c)

🔗

Given

x_{2} = a

and the expression you found in the previous task, which of these describes the solution set for this system?

${[\begin{matrix} 2 a + 3 \\ a \end{matrix}] | a \in R}$
${[\begin{matrix} a \\ 2 a + 3 \end{matrix}] | a \in R}$
${[\begin{matrix} a \\ b \end{matrix}] | a \in R}$
${[\begin{matrix} 2 a + 3 \\ 2 b - 3 \end{matrix}] | a \in R}$

Answer.

{[\begin{matrix} 2 a - 3 \\ a \end{matrix}] | a \in R}

🔗

Activity 1.1.14.

🔗

Consider the following linear system.

\begin{aligned} x_{1} & + & 2 x_{2} & - & x_{4} & = & 3 \\ x_{3} & + & 4 x_{4} & = & - 2 \end{aligned}

🔗

Substitute

x_{2} = a

and

x_{4} = b,

and then solve for

x_{1}

and

x_{3} :

x_{1} = ? x_{3} = ?

🔗

Then use these to describe the solution set

{[\begin{matrix} ? \\ a \\ ? \\ b \end{matrix}] | a, b \in R}

🔗

to the linear system.

🔗

Observation 1.1.15.

🔗

Solving linear systems of two variables by graphing or substitution is reasonable for two-variable systems, but these simple techniques won’t usually cut it for equations with more than two variables or more than two equations. For example,

\begin{aligned} - 2 x_{1} & - & 4 x_{2} & + & x_{3} & - & 4 x_{4} & = & - 8 \\ x_{1} & + & 2 x_{2} & + & 2 x_{3} & + & 12 x_{4} & = & - 1 \\ x_{1} & + & 2 x_{2} & + & x_{3} & + & 8 x_{4} & = & 1 \end{aligned}

🔗

has the exact same solution set as the system in the previous activity, but we’ll want to learn new techniques to compute these solutions efficiently.

🔗

Remark 1.1.16.

🔗

The only important information in a linear system are its coefficients and constants.

🔗

Original linear system:

\begin{aligned} x_{1} + 3 x_{3} & = & 3 \\ 3 x_{1} - 2 x_{2} + 4 x_{3} & = & 0 \\ - x_{2} + x_{3} & = & - 2 \end{aligned}

🔗

Verbose standard form:

\begin{aligned} 1 x_{1} & + & 0 x_{2} & + & 3 x_{3} & = & 3 \\ 3 x_{1} & - & 2 x_{2} & + & 4 x_{3} & = & 0 \\ 0 x_{1} & - & 1 x_{2} & + & 1 x_{3} & = & - 2 \end{aligned}

🔗

Coefficients/constants:

\begin{aligned} 1 & 0 & 3 & | & 3 \\ 3 & - 2 & 4 & | & 0 \\ 0 & - 1 & 1 & | & - 2 \end{aligned}

🔗

Definition 1.1.17.

🔗

A system of

m

linear equations with

n

variables is often represented by writing its coefficients and constants in an augmented matrix: the

m \times n

matrix of its coefficients augmented with the

m

constant values as a final column.

\begin{aligned} a_{11} x_{1} & + & a_{12} x_{2} & + & \dots & + & a_{1 n} x_{n} & = & b_{1} \\ a_{21} x_{1} & + & a_{22} x_{2} & + & \dots & + & a_{2 n} x_{n} & = & b_{2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ a_{m 1} x_{1} & + & a_{m 2} x_{2} & + & \dots & + & a_{m n} x_{n} & = & b_{m} \end{aligned}

[\begin{array}{ccccc} a_{11} & a_{12} & \dots & a_{1 n} & b_{1} \\ a_{21} & a_{22} & \dots & a_{2 n} & b_{2} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} & b_{m} \end{array}]

Sometimes, we will find it useful to refer only to the coefficients of the linear system (and ignore its constant terms). We call the

m \times n

array consisting of these coefficients a coefficient matrix.

[\begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{array}]

🔗

Example 1.1.18.

🔗

The corresponding augmented matrix for this system is obtained by simply writing the coefficients and constants in matrix form.

🔗

Linear system:

\begin{aligned} x_{1} & + & 3 x_{3} & = & 3 \\ 3 x_{1} & - & 2 x_{2} & + & 4 x_{3} & = & 0 \\ - & x_{2} & + & x_{3} & = & - 2 \end{aligned}

🔗

Augmented matrix:

[\begin{array}{cccc} 1 & 0 & 3 & 3 \\ 3 & - 2 & 4 & 0 \\ 0 & - 1 & 1 & - 2 \end{array}]

🔗

Vector equation:

x_{1} [\begin{matrix} 1 \\ 3 \\ 0 \end{matrix}] + x_{2} [\begin{matrix} 0 \\ - 2 \\ - 1 \end{matrix}] + x_{3} [\begin{matrix} 3 \\ 4 \\ 1 \end{matrix}] = [\begin{matrix} 3 \\ 0 \\ - 2 \end{matrix}]

🔗

Subsection 1.1.3 Individual Practice

🔗

Activity 1.1.19.

🔗

Consider the following augmented matrices. For each of them, decide how many variables and how many equations the corresponding linear system has.

🔗

(a)

🔗

[\begin{array}{cccc} 2 & 1 & 3 & 3 \\ 1 & - 2 & 4 & 3 \\ 3 & - 1 & 7 & - 1 \end{array}]

🔗

(b)

🔗

[\begin{array}{cccc} 2 & 1 & 3 & 3 \\ 1 & - 2 & 4 & 3 \\ 3 & - 1 & 7 & - 1 \\ 3 & - 1 & 7 & - 1 \end{array}]

🔗

(c)

🔗

[\begin{array}{cccc} 2 & 0 & 3 & 3 \\ 1 & 0 & 4 & 3 \\ 3 & 0 & 7 & - 1 \\ 3 & 0 & 7 & - 1 \end{array}]

🔗

(d)

🔗

[\begin{array}{cccc} 2 & 1 & 3 & 3 \\ 1 & - 2 & 4 & 3 \\ 0 & 0 & 0 & 0 \\ 3 & - 1 & 7 & - 1 \end{array}]

🔗

Subsection 1.1.4 Videos

🔗

Figure 1. Video: Converting between systems, vector equations, and augmented matrices

🔗

Choose a value for the real constant

k

such that the following system has one, many, or no solutions. In each case, write the solution set.

🔗

Consider the linear system:

\begin{aligned} x_{1} - x_{2} & = & 1 \\ 3 x_{1} - 3 x_{2} & = & k \end{aligned}

🔗

Exploration 1.1.21.

🔗

Consider the linear system:

\begin{aligned} a x_{1} + b x_{2} & = & j \\ c x_{1} + d x_{2} & = & k \end{aligned}

Assume

j

and

k

are arbitrary real numbers.

Choose values for $a, b, c,$ and $d,$ such that $a d - b c = 0 .$ Show that this system is inconsistent.
Prove that, if $a d - b c \neq 0,$ the system is consistent with exactly one solution.

🔗

Exploration 1.1.22.

🔗

Given a set

S,

we can define a relation between two arbitrary elements

a, b \in S .

If the two elements are related, we denote this

a \sim b .

🔗

Any relation on a set

S

that satisfies the properties below is an equivalence relation.

Reflexive: For any $a \in S, a \sim a$
Symmetric: For $a, b \in S,$ if $a \sim b,$ then $b \sim a$
Transitive: for any $a, b, c \in S, a \sim b and b \sim c implies a \sim c$

🔗

For each of the following relations, show that it is or is not an equivalence relation.

For $a, b, \in R,$ $a \sim b$ if an only if $a \leq b .$
For $a, b, \in R,$ $a \sim b$ if an only if $| a | = | b | .$

🔗

Subsection 1.1.7 Sample Problem and Solution

🔗

Sample problem Example B.1.1.

Prev Top Next