Learning Outcomes
- Determine the number of solutions for a system of linear equations or a vector equation.
Section 1.3 Counting Solutions for Linear Systems (LE3)
Subsection 1.3.1 Warm Up
Activity 1.3.1.
(a)
Without referring to your Activity Book, which of the four criteria for a matrix to be in Reduced Row Echelon Form (RREF) can you recall?
(b)
Which, if any, of the following matrices are in RREF? You may refer to the Activity Book now for criteria that you may have forgotten.
\begin{equation*}
P=\left[\begin{array}{ccc|c} 1 & 0 & \frac{2}{3} & -3 \\ 0 & 3 & 3 & -\frac{3}{5} \\ 0 & 0 & 0 & 0 \end{array}\right]
\end{equation*}
\begin{equation*}
Q=\left[\begin{array}{ccc|c} 0 & 1 & 0 & 7 \\ 1 & 0 & 0 & 4 \\ 0 & 0 & 0 & 0 \end{array}\right]
\end{equation*}
\begin{equation*}
R=\left[\begin{array}{ccc|c} 1 & 0 & \frac{1}{2} & 4 \\ 0 & 1 & 0 & 7 \\ 0 & 0 & 1 & 0 \end{array}\right]
\end{equation*}
Subsection 1.3.2 Class Activities
Remark 1.3.2.
We will frequently need to know the reduced row echelon form of matrices during the remainder of this course, so unless you’re told otherwise, feel free to use technology (see Activity 1.2.19) to compute RREFs efficiently.
Activity 1.3.3.
Consider the following system of equations.
\begin{alignat*}{4}
3x_1 &\,-\,& 2x_2 &\,+\,& 13x_3 &\,=\,& 6\\
2x_1 &\,-\,& 2x_2 &\,+\,& 10x_3 &\,=\,& 2\\
-x_1 &\,+\,& 3x_2 &\,-\,& 6x_3 &\,=\,& 11\\
4x_1 &\,+\,& x_2 &\,+\,& x_3 &\,=\,& 1\text{.}
\end{alignat*}
(a)
Convert this to an augmented matrix and use technology to compute its reduced row echelon form:
\begin{equation*}
\RREF
\left[\begin{array}{ccc|c}
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\end{array}\right]
=
\left[\begin{array}{ccc|c}
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\end{array}\right]
\end{equation*}
(b)
Use the \(\RREF\) matrix to write a linear system equivalent to the original system.
(c)
How many solutions must this system have?
- Zero
- Only one
- Infinitely-many
Activity 1.3.4.
Consider the vector equation
\begin{equation*}
x_1 \left[\begin{array}{c} 3 \\ 2\\ -1 \\ 3 \end{array}\right]
+x_2 \left[\begin{array}{c}-2 \\ -2 \\ 0 \\ 7 \end{array}\right]
+x_3\left[\begin{array}{c} 13 \\ 10 \\ -3 \\ 0 \end{array}\right]
=\left[\begin{array}{c} 6 \\ 2 \\ 1 \\ -2 \end{array}\right]
\end{equation*}
(a)
Convert this to an augmented matrix and use technology to compute its reduced row echelon form:
\begin{equation*}
\RREF
\left[\begin{array}{ccc|c}
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\end{array}\right]
=
\left[\begin{array}{ccc|c}
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\unknown&\unknown&\unknown&\unknown\\
\end{array}\right]
\end{equation*}
(b)
Use the \(\RREF\) matrix to write a linear system equivalent to the original system.
(c)
How many solutions must this system have?
- Zero
- Only one
- Infinitely-many
Activity 1.3.5.
What contradictory equations besides \(0=1\) may be obtained from the RREF of an augmented matrix?
- \(x=0\) is an obtainable contradiction
- \(x=y\) is an obtainable contradiction
- \(0=17\) is an obtainable contradiction
- \(0=1\) is the only obtainable contradiction
Activity 1.3.6.
Consider the following linear system.
\begin{alignat*}{4}
x_1 &+ 2x_2 &+ 3x_3 &= 1\\
2x_1 &+ 4x_2 &+ 8x_3 &= 0\\
3x_1 &+ 6x_2 &+ 11x_3 &= 1\\
x_1 &+ 2x_2 &+ 5x_3 &= -1
\end{alignat*}
(a)
Find its corresponding augmented matrix \(A\) and find \(\RREF(A)\text{.}\)
(b)
Use the \(\RREF\) matrix to write a linear system equivalent to the original system.
(c)
How many solutions must this system have?
- Zero
- One
- Infinitely-many
Fact 1.3.7.
By finding \(\RREF(A)\) from a linear system’s corresponding augmented matrix \(A\text{,}\) we can immediately tell how many solutions the system has.
- If the linear system given by \(\RREF(A)\) includes the contradiction\begin{equation*} 0=1\text{,} \end{equation*}that is, the \(\RREF\) matrix includes the row\begin{equation*} \left[\begin{array}{ccc|c}0&\cdots&0&1\end{array}\right]\text{,} \end{equation*}then the system is inconsistent, which means it has zero solutions and we may write\begin{equation*} \text{Solution set }=\{\} \hspace{2em}\text{or}\hspace{2em} \text{Solution set }=\emptyset. \end{equation*}
- If the linear system given by \(\RREF(A)\) sets each variable of the system to a single value; that is we have:\begin{alignat*}{2} x_1 &= s_1\\ x_2 &= s_2\\ &\vdots\\ x_n &= s_n \end{alignat*}(with some possible extra \(0=0\) equations), then the system is consistent with exactly one solution, and we may write\begin{equation*} \text{Solution }=\left[\begin{array}{c}s_1\\s_2\\\vdots\\s_n\end{array}\right] \hspace{2em}\text{but}\hspace{2em} \text{Solution set }= \left\{\left[\begin{array}{c}s_1\\s_2\\\vdots\\s_n\end{array}\right]\right\}. \end{equation*}
- Otherwise, the system given by the \(\RREF\) matrix must not include a \(0=1\) contradiction while including at least one equation with multiple variables. This means it is consistent with infinitely-many different solutions. We’ll learn how to find such solution sets in Section 1.4.
Activity 1.3.8.
Consider each of the following systems of linear equations or vector equations.
(a)
\begin{equation*}
\begin{matrix} x_{1} & - & x_{2} & - & 3 \, x_{3} & = & 8 \\ 3 \, x_{1} & - & 2 \, x_{2} & - & 5 \, x_{3} & = & 17 \\ x_{1} & - & x_{2} & - & 2 \, x_{3} & = & 7 \\ 10 \, x_{1} & - & 8 \, x_{2} & - & 21 \, x_{3} & = & 65 \\ \end{matrix}
\end{equation*}
(i)
Explain and demonstrate how to find a simpler linear system that has the same solution set.
Answer.
\begin{equation*}
\begin{matrix} x_{1} & & & & & = & 2 \\ & & x_{2} & & & = & -3 \\ & & & & x_{3} & = & -1 \\ & & & & 0 & = & 0 \\ \end{matrix}
\end{equation*}
(ii)
Explain whether this solution set has no solutions, one solution, or infinitely-many solutions. If the set is finite, describe it using set notation.
Answer.
The solution set has one solution. The solution set is \(\left\{ \left[\begin{array}{c} 2 \\ -3 \\ -1 \end{array}\right] \right\}\text{.}\)
(b)
\begin{equation*}
\begin{matrix} x_{1} & - & 5 \, x_{2} & - & 15 \, x_{3} & = & -8 \\ & & x_{2} & + & 3 \, x_{3} & = & 1 \\ x_{1} & & & & & = & 2 \\ 5 \, x_{1} & - & 7 \, x_{2} & - & 21 \, x_{3} & = & -10 \\ \end{matrix}
\end{equation*}
(i)
Explain and demonstrate how to find a simpler linear system that has the same solution set.
Answer.
\begin{equation*}
\begin{matrix} x_{1} & & & & & = & 0 \\ & & x_{2} & + & 3 \, x_{3} & = & 0 \\ & & & & 0 & = & 1 \\ & & & & 0 & = & 0 \\ \end{matrix}
\end{equation*}
(ii)
Explain whether this solution set has no solutions, one solution, or infinitely-many solutions. If the set is finite, describe it using set notation.
Answer.
The solution set has no solutions. The solution set is \(\emptyset\text{.}\)
(c)
\begin{equation*}
\begin{matrix} -2 \, x_{1} & + & 2 \, x_{2} & + & 5 \, x_{3} & = & 1 \\ -x_{1} & + & x_{2} & + & 2 \, x_{3} & = & 1 \\ 2 \, x_{1} & - & 2 \, x_{2} & + & x_{3} & = & -7 \\ -2 \, x_{1} & + & 2 \, x_{2} & + & 16 \, x_{3} & = & -10 \\ \end{matrix}
\end{equation*}
(i)
Explain and demonstrate how to find a simpler linear system that has the same solution set.
Answer.
\begin{equation*}
\begin{matrix} x_{1} & - & x_{2} & & & = & -3 \\ & & & & x_{3} & = & -1 \\ & & & & 0 & = & 0 \\ & & & & 0 & = & 0 \\ \end{matrix}
\end{equation*}
(ii)
Explain whether this solution set has no solutions, one solution, or infinitely-many solutions. If the set is finite, describe it using set notation.
Answer.
The solution set has infinitely-many solutions.
Subsection 1.3.3 Individual Practice
Activity 1.3.9.
In Fact 1.1.11, we stated, but did not prove the assertion that 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 \(\emptyset\text{.}\)
Explain why this fact is a consequence of Fact 1.3.7 above.
Subsection 1.3.4 Videos
Exercises 1.3.5 Exercises
Exercises available at
https://tbil.org/linear-algebra/preview/exercises/#/bank/LE3/.Subsection 1.3.6 Mathematical Writing Explorations
Exploration 1.3.10.
A system of equations with all constants equal to 0 is called homogeneous. These are addressed in detail in section Section 2.7
- Choose three systems of equations from this chapter that you have already solved. Replace the constants with 0 to make the systems homogeneous. Solve the homogeneous systems and make a conjecture about the relationship between the earlier solutions you found and the associated homogeneous systems.
- Prove or disprove. A system of linear equations is homogeneous if an only if it has the the zero vector as a solution.
Subsection 1.3.7 Sample Problem and Solution
Sample problem Example B.1.3.
