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In order to successfully complete this assignment, you must do the required reading, watch the provided videos, and complete all instructions. The embedded survey form must be entirely filled out and submitted on or before 11:59pm on the day before class. Students must come to class the next day prepared to discuss the material covered in this assignment.

Pre-Class Assignment: Python Linear Algebra Packages

If you have trouble with this assignment a detailed overview of the Gauss-Jordan Elimination method can be found in the Reduced Row Echelon Form section of the Beezer text (Section RREF pg 22-44)

Assignment Overview

  1. The Syntax for Systems of Linear Equations
  2. Introduction to Gauss Jordan Elimination
  3. Gauss Jordan Elimination and the Row Echelon Form
  4. Gauss Jordan Practice
  5. Assignment wrap up

1. The Syntax for Systems of Linear Equations

The following video explains the different syntax we use to describe linear systems.

The following is a summary of the syntax shown in the video:

Linear Equation $$b = a_1x_1+a_2x_2+a_3x_3 + \ldots a_nx_n$$

System of linear equations

$$b_1 = a_{11}x_1+a_{12}x_2+a_{13}x_3 + \ldots a_{1n}$$$$b_2 = a_{21}x_1+a_{22}x_2+a_{23}x_3 + \ldots a_{2n}$$$$b_3 = a_{31}x_1+a_{32}x_2+a_{33}x_3 + \ldots a_{3n}$$$$\vdots$$$$b_m = a_{m1}x_1+a_{m2}x_2+a_{m3}x_3 + \ldots a_{mn}$$

System of linear equations (Matrix format)

$$ \left[ \begin{matrix} b_1 \\ b_2 \\ b_3 \\ \vdots \\ b_m \end{matrix} \right] = \left[ \begin{matrix} a_{11} & a_{12} & a_{13} & & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2n} \\ a_{31} & a_{32} & a_{33} & & a_{3n} \\ & \vdots & & \ddots & \vdots \\ a_{m1} & a_{m2} & a_{m3} & & a_{mn} \end{matrix} \right] \left[ \begin{matrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_m \end{matrix} \right] $$$$b=Ax$$

System of linear equations (Augmented Form)

$$ \left[ \begin{matrix} a_{11} & a_{12} & a_{13} & & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2n} \\ a_{31} & a_{32} & a_{33} & & a_{3n} \\ & \vdots & & \ddots & \vdots \\ a_{m1} & a_{m2} & a_{m3} & & a_{mn} \end{matrix} \, \middle\vert \, \begin{matrix} b_1 \\ b_2 \\ b_3 \\ \vdots \\ b_m \end{matrix} \right] $$

1. Introduction to Gauss Jordan Elimination

The following elementary row operations

  1. Interchange two rows of a matrix
  2. Multiply the elements of a row by a nonzero constant
  3. Add a multiple of the elements of one row to the corresponding elements of another

Consider the element $a_{2,1}$ in the following $A$ Matrix.
$$ A = \left[ \begin{matrix} 1 & 1 \\ 20 & 25 \end{matrix} \, \middle\vert \, \begin{matrix} 30 \\ 690 \end{matrix} \right] $$

✅ QUESTION: : Describe an elementary row operation that could be used to make element $a_{(2,1)}$ zero?

Put your answer here.

✅ QUESTION: : What is the new matrix given the above row operation.

Modify the contents of this cell and put your answer to the above question here.
$$ A = \left[ \begin{matrix} 1 & 1 \\ 0 & ?? \end{matrix} \, \middle\vert \, \begin{matrix} 30 \\ ?? \end{matrix} \right] $$

Hint, we are using a formating language called Latex to display the above matrix. You should just be able to replace the ?? with your new numbers. If you can't figure out what is going on, try searching the web with "latex math and matrix." If it still doesn't make sense, format your answer in another way that will be clear to understand by the you and the instructor.

The following function is a basic implementation of the Gauss-Jorden algorithm to an (m,m+1) augmented matrix:

2. Gauss Jordan Elimination and the Row Echelon Form

The above video left out a special case for Reduced Row Echelon form. There can be non-zero elements in columns that do not have a leading one. For example, All of the following are in Reduced Row Echelon form:

$$ \left[ \begin{matrix} 1 & 2 & 0 & 3 & 0 & 4 \\ 0 & 0 & 1 & 2 & 0 & 7 \\ 0 & 0 & 0 & 0 & 1 & 6 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right] $$$$ \left[ \begin{matrix} 1 & 2 & 0 & 0 & 4 \\ 0 & 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 1 & 5 \end{matrix} \right] $$

✅ QUESTION: : What are the three steps in the Gauss-Jordan Elimination algorithm?

Put your answer here.

3. Gauss Jordan Practice

✅ DO THIS:: Solve the following system of linear equations using the Gauss-Jordan algorithm. Try to do this before watching the video!

$$x_1 + x_3 = 3$$$$2x_2 - 2x_3 = -4$$$$x_2 - 2x_3 = 5$$

Put your answer here

In the following video, we solve the same set of linear equations. Watch the video after trying to do this on your own. It is provided here in case you get stuck.

✅ QUESTION:: Something was unclear in the above videos. Describe the difference between a matrix in "row echelon" form and "reduced row echelon" form.

Put your answer to the above question here

4. Assignment wrap up

Please fill out the form that appears when you run the code below. You must completely fill this out in order to receive credit for the assignment!

Direct Link to Google Form

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✅ Assignment-Specific QUESTION: Describe the difference between a matrix in "row echelon" form and "reduced row echelon" form.

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✅ QUESTION: Summarize what you did in this assignment.

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✅ QUESTION: What questions do you have, if any, about any of the topics discussed in this assignment after working through the jupyter notebook?

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✅ QUESTION: How well do you feel this assignment helped you to achieve a better understanding of the above mentioned topic(s)?

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✅ QUESTION: What was the most challenging part of this assignment for you?

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✅ QUESTION: Do you have any further questions or comments about this material, or anything else that's going on in class?

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✅ QUESTION: Approximately how long did this pre-class assignment take?

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Congratulations, we're done!

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Written by Dr. Dirk Colbry, Michigan State University Creative Commons License
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