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## Section1.1Linear Systems, Vector Equations, and Augmented Matrices (LE1)

### Subsection1.1.1Class Activities

#### Definition1.1.1.

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 that can be plugged into the equation).

#### Remark1.1.2.

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.

#### Definition1.1.3.

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*}

#### Remark1.1.4.

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*}

#### Remark1.1.5.

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*}

#### Definition1.1.6.

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.

#### Activity1.1.8.

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*}

#### Activity1.1.9.

Consider the following consistent linear system.

\begin{align*} -x_1+2x_2 &= -3\\ 2x_1-4x_2 &= 6 \end{align*}
##### (a)

Find three different solutions for this system.

##### (b)

Let $$x_2=a$$ where $$a$$ is an arbitrary real number, then find an expression for $$x_1$$ in terms of $$a\text{.}$$ Use this to write the solution set $$\setBuilder { \left[\begin{array}{c} \unknown \\ a \end{array}\right] }{ a \in \IR }$$ for the linear system.

#### Activity1.1.10.

Consider the following linear system.

\begin{alignat*}{5} x_1 &\,+\,& 2x_2 &\, \,& &\,-\,& x_4 &\,=\,& 3\\ &\, \,& &\, \,& x_3 &\,+\,& 4x_4 &\,=\,& -2 \end{alignat*}

Describe the solution set

\begin{equation*} \setBuilder { \left[\begin{array}{c} \unknown \\ a \\ \unknown \\ b \end{array}\right] }{ a,b \in \IR } \end{equation*}

to the linear system by setting $$x_2=a$$ and $$x_4=b\text{,}$$ and then solving for $$x_1$$ and $$x_3\text{.}$$

#### Observation1.1.11.

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.

#### Remark1.1.12.

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*}

#### Definition1.1.13.

A system of $$m$$ linear equations with $$n$$ variables is often represented by writing its coefficients and constants in an augmented matrix.

\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*}

#### Example1.1.14.

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

Linear system:

\begin{alignat*}{2} 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*}

### Subsection1.1.3Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/LE1.slides.html.

### Subsection1.1.5Mathematical Writing Explorations

#### Exploration1.1.15.

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*}

#### Exploration1.1.16.

Consider the linear system:

\begin{alignat*}{2} ax_1 + bx_2 &\,=\,& j\\ cx_1 + dx_2 &\,=\,& k \end{alignat*}
Assume $$j$$ and $$k$$ are arbitrary real numbers.

• 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.

#### Exploration1.1.17.

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 $$|a|=|b|\text{.}$$

### Subsection1.1.6Sample Problem and Solution

Sample problem Example B.1.1.

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