Learning Outcomes
- Translate back and forth between a system of linear equations, a vector equation, and the corresponding augmented matrix.
Section 1.1 Linear Systems, Vector Equations, and Augmented Matrices (LE1)
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{align*}
-x_1+3x_2 &= 1\\
2x_1-5x_2 &= 2
\end{align*}
(b)
\begin{align*}
-x_1+3x_2 &= 1\\
2x_1-6x_2 &= -2
\end{align*}
(c)
\begin{align*}
-x_1+3x_2 &= 1\\
2x_1-6x_2 &= 3
\end{align*}
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{equation*}
\left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{array}\right]
=
\left[\begin{array}{cccc} \vec v_1 & \vec v_2 & \cdots & \vec v_n\end{array}\right]\text{.}
\end{equation*}
Frequently we will use matrices to describe an ordered list of its column vectors:
\begin{equation*}
\left[\begin{array}{c}
a_{11} \\
a_{21} \\
\vdots \\
a_{m1} \\
\end{array}\right],
\left[\begin{array}{c}
a_{12} \\
a_{22} \\
\vdots \\
a_{m2} \\
\end{array}\right],\cdots,
\left[\begin{array}{c}
a_{1n} \\
a_{2n} \\
\vdots \\
a_{mn} \\
\end{array}\right] =
\vec v_1, \vec v_2, \cdots, \vec v_n\text{.}
\end{equation*}
When order is irrelevant, we will use set notation:
\begin{equation*}
\left\{
\left[\begin{array}{c}
a_{11} \\
a_{21} \\
\vdots \\
a_{m1} \\
\end{array}\right],
\left[\begin{array}{c}
a_{12} \\
a_{22} \\
\vdots \\
a_{m2} \\
\end{array}\right],\cdots,
\left[\begin{array}{c}
a_{1n} \\
a_{2n} \\
\vdots \\
a_{mn} \\
\end{array}\right]\right\} =
\{\vec v_1, \vec v_2, \cdots, \vec v_n\}\text{.}
\end{equation*}
Definition 1.1.3.
A Euclidean vector is an ordered list of real numbers
\begin{equation*}
\left[\begin{array}{c}
a_1 \\
a_2 \\
\vdots \\
a_n
\end{array}\right]\text{.}
\end{equation*}
We will find it useful to almost always typeset Euclidean vectors vertically, but the notation \(\left[\begin{array}{cccc}a_1 & a_2 & \cdots & a_n\end{array}\right]^T\) is also valid when vertical typesetting is inconvenient. The set of all Euclidean vectors with \(n\) components is denoted as \(\mathbb R^n\text{,}\) and vectors are often described using the notation \(\vec v\text{.}\)
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{equation*}
\left[\begin{array}{c}
a_1 \\
a_2 \\
\vdots \\
a_n
\end{array}\right]+
\left[\begin{array}{c}
b_1 \\
b_2 \\
\vdots \\
b_n
\end{array}\right]=
\left[\begin{array}{c}
a_1+b_1 \\
a_2+b_2 \\
\vdots \\
a_n+b_n
\end{array}\right]
\hspace{3em}
c
\left[\begin{array}{c}
a_1 \\
a_2 \\
\vdots \\
a_n
\end{array}\right]=
\left[\begin{array}{c}
ca_1 \\
ca_2 \\
\vdots \\
ca_n
\end{array}\right]
\end{equation*}
Example 1.1.4.
Following are some examples of addition and scalar multiplication in \(\mathbb R^4\text{.}\)
\begin{equation*}
\left[\begin{array}{c}
3 \\
-3 \\
0 \\
4
\end{array}\right]+
\left[\begin{array}{c}
0 \\
2 \\
7 \\
1
\end{array}\right]=
\left[\begin{array}{c}
3+0 \\
-3+2 \\
0+7 \\
4+1
\end{array}\right]=
\left[\begin{array}{c}
3 \\
-1 \\
7 \\
5
\end{array}\right]
\end{equation*}
\begin{equation*}
-4
\left[\begin{array}{c}
0 \\
2 \\
-2 \\
3
\end{array}\right]=
\left[\begin{array}{c}
-4(0) \\
-4(2)\\
-4(-2) \\
-4(3)
\end{array}\right]=
\left[\begin{array}{c}
0 \\
-8 \\
8 \\
-12
\end{array}\right]
\end{equation*}
Definition 1.1.5.
A linear equation is an equation of the variables \(x_i\) of the form
\begin{equation*}
a_1x_1+a_2x_2+\dots+a_nx_n=b\text{.}
\end{equation*}
A solution for a linear equation is a Euclidean vector
\begin{equation*}
\left[\begin{array}{c}
s_1 \\
s_2 \\
\vdots \\
s_n
\end{array}\right]
\end{equation*}
that satisfies
\begin{equation*}
a_1s_1+a_2s_2+\dots+a_ns_n=b
\end{equation*}
(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 \(x_i\text{,}\) 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{alignat*}{5}
a_{11}x_1 &\,+\,& a_{12}x_2 &\,+\,& \dots &\,+\,& a_{1n}x_n &\,=\,& b_1 \\
a_{21}x_1 &\,+\,& a_{22}x_2 &\,+\,& \dots &\,+\,& a_{2n}x_n &\,=\,& b_2\\
\vdots& &\vdots& && &\vdots&&\vdots \\
a_{m1}x_1 &\,+\,& a_{m2}x_2 &\,+\,& \dots &\,+\,& a_{mn}x_n &\,=\,& b_m
\end{alignat*}
Its solution set is given by
\begin{equation*}
\setBuilder
{
\left[\begin{array}{c}
s_1 \\
s_2 \\
\vdots \\
s_n
\end{array}\right]
}{
\left[\begin{array}{c}
s_1 \\
s_2 \\
\vdots \\
s_n
\end{array}\right]
\text{is a solution to all equations in the system}
}\text{.}
\end{equation*}
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{alignat*}{2}
x_1 + 3x_3 &\,=\,& 3\\
3x_1 - 2x_2 + 4x_3 &\,=\,& 0\\
-x_2 + x_3 &\,=\,& -2
\end{alignat*}
Verbose standard form:
\begin{alignat*}{4}
1x_1 &\,+\,& 0x_2 &\,+\,& 3x_3 &\,=\,& 3\\
3x_1 &\,-\,& 2x_2 &\,+\,& 4x_3 &\,=\,& 0\\
0x_1 &\,-\,& 1x_2 &\,+\,& 1x_3 &\,=\,& -2
\end{alignat*}
Concise standard form:
\begin{alignat*}{4}
x_1 & & &\,+\,& 3x_3 &\,=\,& 3\\
3x_1 &\,-\,& 2x_2 &\,+\,& 4x_3 &\,=\,& 0\\
&\,-\,& x_2 &\,+\,& x_3 &\,=\,& -2
\end{alignat*}
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
\begin{equation*}
x_1 \left[\begin{array}{c} 1 \\ 3 \\ 0 \end{array}\right]+ x_2 \left[\begin{array}{c} 0 \\ -2 \\ -1 \end{array}\right]
+ x_3 \left[\begin{array}{c} 3 \\ 4 \\1 \end{array}\right] = \left[\begin{array}{c} 3 \\ 0 \\ -2 \end{array}\right]
\end{equation*}
is equivalent to the system of equations
\begin{alignat*}{4}
x_1 & & &\,+\,& 3x_3 &\,=\,& 3\\
3x_1 &\,-\,& 2x_2 &\,+\,& 4x_3 &\,=\,& 0\\
&\,-\,& x_2 &\,+\,& x_3 &\,=\,& -2
\end{alignat*}
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. \(\setList{\left[\begin{array}{c}1\\2\\3\end{array}\right]}\)
- Consistent with infinitely-many solutions: its solution set contains infinitely many vectors, e.g. \(\setBuilder { \left[\begin{array}{c}1\\2-3a\\a\end{array}\right] }{ a\in\IR }\)
- Inconsistent: its solution set is the empty set, denoted by either \(\{\}\) or \(\emptyset\text{.}\)
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{align*}
-x_1+2x_2 &= 5\\
2x_1-4x_2 &= 6
\end{align*}
Activity 1.1.13.
Consider the following consistent linear system.
\begin{align*}
-x_1+2x_2 &= -3\\
2x_1-4x_2 &= 6
\end{align*}
(a)
Find several different solutions for this system:
\begin{equation*}
\left[\begin{array}{c}
1 \\
-1
\end{array}\right] \hspace{3em}
\left[\begin{array}{c}
\unknown \\
2
\end{array}\right] \hspace{3em}
\left[\begin{array}{c}
0 \\
\unknown
\end{array}\right] \hspace{3em}
\left[\begin{array}{c}
\unknown \\
\unknown
\end{array}\right] \hspace{3em}
\left[\begin{array}{c}
\unknown \\
\unknown
\end{array}\right]
\end{equation*}
(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?
- \(\displaystyle x_1=-3a\)
- \(\displaystyle x_1=3\)
- \(\displaystyle x_1=2a-3\)
- \(\displaystyle x_1=-4a+6\)
Answer.
C. \(x_1=2a-3\)
(c)
Given \(x_2=a\) and the expression you found in the previous task, which of these describes the solution set for this system?
- \(\displaystyle \setBuilder { \left[\begin{array}{c} 2a-3 \\ a \end{array}\right] }{ a \in \IR }\)
- \(\displaystyle \setBuilder { \left[\begin{array}{c} a \\ 2a-3 \end{array}\right] }{ a \in \IR }\)
- \(\displaystyle \setBuilder { \left[\begin{array}{c} a \\ b \end{array}\right] }{ a \in \IR }\)
- \(\displaystyle \setBuilder { \left[\begin{array}{c} 2a-3 \\ 2b-3 \end{array}\right] }{ a \in \IR }\)
Answer.
A. \(\setBuilder
{
\left[\begin{array}{c}
2a-3 \\
a
\end{array}\right]
}{
a \in \IR
}\)
Activity 1.1.14.
Consider the following linear system.
\begin{alignat*}{5}
x_1 &\,+\,& 2x_2 &\, \,& &\,-\,& x_4 &\,=\,& 3\\
&\, \,& &\, \,& x_3 &\,+\,& 4x_4 &\,=\,& -2
\end{alignat*}
Substitute \(x_2=a\) and \(x_4=b\text{,}\) and then solve for \(x_1\) and \(x_3\text{:}\)
\begin{equation*}
x_1 = \unknown \hspace{6em} x_3 = \unknown \hspace{6em}
\end{equation*}
Then use these to describe the solution set
\begin{equation*}
\setBuilder
{
\left[\begin{array}{c}
\hspace{3em}\unknown\hspace{3em} \\
a \\
\unknown \\
b
\end{array}\right]
}{
a,b \in \IR
}
\end{equation*}
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{alignat*}{5}
-2x_1 &\,-\,& 4x_2 &\,+\,& x_3 &\,-\,& 4x_4 &\,=\,& -8\\
x_1 &\,+\,& 2x_2 &\,+\,& 2x_3 &\,+\,& 12x_4 &\,=\,& -1\\
x_1 &\,+\,& 2x_2 &\,+\,& x_3 &\,+\,& 8x_4 &\,=\,& 1
\end{alignat*}
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{alignat*}{2}
x_1 + 3x_3 &\,=\,& 3\\
3x_1 - 2x_2 + 4x_3 &\,=\,& 0\\
-x_2 + x_3 &\,=\,& -2
\end{alignat*}
Verbose standard form:
\begin{alignat*}{4}
1x_1 &\,+\,& 0x_2 &\,+\,& 3x_3 &\,=\,& 3\\
3x_1 &\,-\,& 2x_2 &\,+\,& 4x_3 &\,=\,& 0\\
0x_1 &\,-\,& 1x_2 &\,+\,& 1x_3 &\,=\,& -2
\end{alignat*}
Coefficients/constants:
\begin{alignat*}{4}
1 & & 0 &\,\,& 3 &\,|\,& 3\\
3 &\, \,& -2 &\,\,& 4 &\,|\,& 0\\
0 &\, \,& -1 &\,\,& 1 &\,|\,& -2
\end{alignat*}
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{alignat*}{5}
a_{11}x_1 &\,+\,& a_{12}x_2 &\,+\,& \dots &\,+\,& a_{1n}x_n &\,=\,& b_1\\
a_{21}x_1 &\,+\,& a_{22}x_2 &\,+\,& \dots &\,+\,& a_{2n}x_n &\,=\,& b_2\\
\vdots& &\vdots& && &\vdots&&\vdots\\
a_{m1}x_1 &\,+\,& a_{m2}x_2 &\,+\,& \dots &\,+\,& a_{mn}x_n &\,=\,& b_m
\end{alignat*}
\begin{equation*}
\left[\begin{array}{cccc|c}
a_{11} & a_{12} & \cdots & a_{1n} & b_1\\
a_{21} & a_{22} & \cdots & a_{2n} & b_2\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn} & b_m
\end{array}\right]
\end{equation*}
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{equation*}
\left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n}\\
a_{21} & a_{22} & \cdots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{array}\right]
\end{equation*}
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{alignat*}{4}
x_1 && &\,+\,& 3x_3 &\,=\,& 3\\
3x_1 &\,-\,& 2x_2 &\,+\,& 4x_3 &\,=\,& 0\\
&\,-\,& x_2 &\,+\,& x_3 &\,=\,& -2
\end{alignat*}
Augmented matrix:
\begin{equation*}
\left[\begin{array}{ccc|c}
1 & 0 & 3 & 3 \\
3 & -2 & 4 & 0 \\
0 & -1 & 1 & -2
\end{array}\right]
\end{equation*}
Vector equation:
\begin{equation*}
x_1 \left[\begin{array}{c} 1 \\ 3 \\ 0 \end{array}\right]+ x_2 \left[\begin{array}{c} 0 \\ -2 \\ -1 \end{array}\right] + x_3 \left[\begin{array}{c} 3 \\ 4 \\1 \end{array}\right] = \left[\begin{array}{c} 3 \\ 0 \\ -2 \end{array}\right]
\end{equation*}
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{equation*}
\left[\begin{array}{ccc|c}
2 & 1 & 3 & 3 \\
1 & -2 & 4 & 3 \\
3 & -1 & 7 & -1
\end{array}\right]
\end{equation*}
(b)
\begin{equation*}
\left[\begin{array}{ccc|c}
2 & 1 & 3 & 3 \\
1 & -2 & 4 & 3 \\
3 & -1 & 7 & -1 \\
3 & -1 & 7 & -1
\end{array}\right]
\end{equation*}
(c)
\begin{equation*}
\left[\begin{array}{ccc|c}
2 & 0 & 3 & 3 \\
1 & 0 & 4 & 3 \\
3 & 0 & 7 & -1 \\
3 & 0 & 7 & -1
\end{array}\right]
\end{equation*}
(d)
\begin{equation*}
\left[\begin{array}{ccc|c}
2 & 1 & 3 & 3 \\
1 & -2 & 4 & 3 \\
0 & 0 & 0 & 0 \\
3 & -1 & 7 & -1
\end{array}\right]
\end{equation*}
Subsection 1.1.4 Videos
Exercises 1.1.5 Exercises
Exercises available at
https://tbil.org/linear-algebra/preview/exercises/#/bank/LE1/.Subsection 1.1.6 Mathematical Writing Explorations
Exploration 1.1.20.
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{alignat*}{2}
x_1 - x_2 &\,=\,& 1\\
3x_1 - 3x_2 &\,=\,& k
\end{alignat*}
Exploration 1.1.21.
Consider the linear system:
\begin{alignat*}{2}
ax_1 + bx_2 &\,=\,& j\\
cx_1 + dx_2 &\,=\,& k
\end{alignat*}
- Choose values for \(a,b,c\text{,}\) and \(d\text{,}\) such that \(ad-bc = 0\text{.}\) Show that this system is inconsistent.
- Prove that, if \(ad-bc \neq 0\text{,}\) the system is consistent with exactly one solution.
Exploration 1.1.22.
Given a set \(S\text{,}\) we can define a relation between two arbitrary elements \(a,b \in S\text{.}\) If the two elements are related, we denote this \(a \sim b\text{.}\)
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 \(a,b \in S\text{,}\) if \(a\sim b\text{,}\) then \(b \sim a\)
- Transitive: for any \(a,b,c \in S, a \sim b \mbox{ and } b \sim c \mbox{ implies } a\sim c\)
For each of the following relations, show that it is or is not an equivalence relation.
- For \(a,b, \in \mathbb{R}\text{,}\) \(a \sim b\) if an only if \(a \leq b\text{.}\)
- For \(a,b, \in \mathbb{R}\text{,}\) \(a \sim b\) if an only if \(|a|=|b|\text{.}\)
Subsection 1.1.7 Sample Problem and Solution
Sample problem Example B.1.1.
