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Section 1.3 Counting Solutions for Linear Systems (LE3)

Subsection 1.3.1 Class Activities

Activity 1.3.1.

Free browser-based technologies for mathematical computation are available online.

  • Go to https://sagecell.sagemath.org/.

  • In the dropdown on the right, you can select a number of different languages. Select "Octave" for the Matlab-compatible syntax used by this text.

  • Type rref([1,3,2;2,5,7]) and then press the Evaluate button to compute the \(\RREF\) of \(\left[\begin{array}{ccc} 1 & 3 & 2 \\ 2 & 5 & 7 \end{array}\right]\text{.}\)

Since the vertical bar in an augmented matrix does not affect row operations, the \(\RREF\) of \(\left[\begin{array}{cc|c} 1 & 3 & 2 \\ 2 & 5 & 7 \end{array}\right]\) may be computed in the same way.

Activity 1.3.2.

In the HTML version of this text, code cells are often embedded for your convenience when RREFs need to be computed.

Try this out to compute \(\RREF\left[\begin{array}{cc|c} 2 & 3 & 1 \\ 3 & 0 & 6 \end{array}\right]\text{.}\)

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\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\\ \end{array}\right] = \left[\begin{array}{ccc|c} \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?

  1. Zero

  2. Only one

  3. Infinitely-many

Activity 1.3.4.

Consider the vector equation

\begin{equation*} x_1 \left[\begin{array}{c} 3 \\ 2\\ -1 \end{array}\right] +x_2 \left[\begin{array}{c}-2 \\ -2 \\ 0 \end{array}\right] +x_3\left[\begin{array}{c} 13 \\ 10 \\ -3 \end{array}\right] =\left[\begin{array}{c} 6 \\ 2 \\ 1 \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\\ \end{array}\right] = \left[\begin{array}{ccc|c} \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?

  1. Zero

  2. Only one

  3. Infinitely-many

Activity 1.3.5.

Is \(0=1\) the only possible logical contradiction obtained from the RREF of an augmented matrix?

  1. Yes, \(0=1\) is the only possible contradiction from an RREF matrix.

  2. No, \(0=17\) is another possible contradiction from an RREF matrix.

  3. No, \(x=0\) is another possible contradiction from an RREF matrix.

  4. No, \(x=y\) is another possible contradiction from an RREF matrix.

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 \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?

  1. Zero

  2. One

  3. Infinitely-many

Activity 1.3.8.

For each vector equation, write an explanation for whether each solution set has no solutions, one solution, or infinitely-many solutions. If the set is finite, describe it using set notation.

(a)
\begin{equation*} x_{1} \left[\begin{array}{c} 1 \\ -1 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -3 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 7 \\ -6 \\ 4 \end{array}\right] = \left[\begin{array}{c} 10 \\ -6 \\ 4 \end{array}\right] \end{equation*}
(b)
\begin{equation*} x_{1} \left[\begin{array}{c} -2 \\ -1 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ 1 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -2 \\ -5 \end{array}\right] = \left[\begin{array}{c} 1 \\ 4 \\ 13 \end{array}\right] \end{equation*}
(c)
\begin{equation*} x_{1} \left[\begin{array}{c} -1 \\ -2 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -5 \\ 4 \end{array}\right] + x_{3} \left[\begin{array}{c} -7 \\ -9 \\ 6 \end{array}\right] = \left[\begin{array}{c} 3 \\ 1 \\ -2 \end{array}\right] \end{equation*}

Subsection 1.3.2 Videos

Figure 3. Video: Finding the number of solutions for a system

Subsection 1.3.3 Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/LE3.slides.html.

Exercises 1.3.4 Exercises

Exercises available at https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/LE3/.

Subsection 1.3.5 Mathematical Writing Explorations

Exploration 1.3.9.

A system of equations with all constants equal to 0 is called homogeneous. These are addressed in detail in section Section 2.9

  • 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 1.3.6 Sample Problem and Solution

Sample problem Example B.1.3.