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.
# Load Useful Python Libraries
%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True)
1. Introduction to Gauss Jordan Elimination¶
The following elementary row operations
- Interchange two rows of a matrix
- Multiply the elements of a row by a nonzero constant
- Add a multiple of the elements of one row to the corresponding elements of another
from IPython.display import YouTubeVideo
YouTubeVideo("iGmtmF_hm2g",width=640,height=360, cc_load_policy=True)
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:
from IPython.display import YouTubeVideo
YouTubeVideo("k5fdGS5b4OU",width=640,height=360, cc_load_policy=True)
Consider the Giselle example from above. Her earnings do not change (i.e. she makes $\$20$ per hour as a carpenter and $\$25$ per hour as a blacksmith). However, now she has worked two more weeks. In the second week, she worked for a total of 35 hours and earned $\$750$. In the third week, she worked for a total of 30 hours and earned $\$650$. How much did she work as a carpenter and blacksmith for each of those weeks? In other words:
Week 1: $$ c + b = 30 $$ $$ 20c + 25b = 690 $$
Week 2: $$ c + b = 35 $$ $$ 20c + 25b = 750 $$
Week 3: $$ c + b = 30 $$ $$ 20c + 25b = 650 $$
✅**DO THIS**: Write a $2 \times 5$ augmented matrix representing the 6 equations above. Name your Matrix $G$ to verify your answer using the checkanswer function below.
#Put your answer to the above quation here
from answercheck import checkanswer
checkanswer.matrix(G,'a1e01de142199370be70131849fbf108');
from IPython.display import YouTubeVideo
YouTubeVideo("v6RstFsrTJY",width=640,height=360, cc_load_policy=True)
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 vidoe after trying to do this on your own. It is provided here in case you get stuck.
from IPython.display import YouTubeVideo
YouTubeVideo("xT16yIVw_KE",width=640,height=360, cc_load_policy=True)
✅ **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
5. 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!
If you have trouble with the embedded form, please make sure you log on with your MSU google account at googleapps.msu.edu and then click on the direct link above.
✅ **Assignment-Specific QUESTION:** Describe the difference between a matrix in "row echelon" form and "reduced row echelon" form.
Put your answer to the above question here
✅ **QUESTION:** Summarize what you did in this assignment.
Put your answer to the above question here
✅ **QUESTION:** What questions do you have, if any, about any of the topics discussed in this assignment after working through the jupyter notebook?
Put your answer to the above question here
✅ **QUESTION:** How well do you feel this assignment helped you to achieve a better understanding of the above mentioned topic(s)?
Put your answer to the above question here
✅ **QUESTION:** What was the most challenging part of this assignment for you?
Put your answer to the above question here
✅ **QUESTION:** What was the least challenging part of this assignment for you?
Put your answer to the above question here
✅ **QUESTION:** What kind of additional questions or support, if any, do you feel you need to have a better understanding of the content in this assignment?
Put your answer to the above question here
✅ **QUESTION:** Do you have any further questions or comments about this material, or anything else that's going on in class?
Put your answer to the above question here
✅ **QUESTION:** Approximately how long did this pre-class assignment take?
Put your answer to the above question here
from IPython.display import HTML
HTML(
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)
Congratulations, we're done!¶
To get credits for this assignment, you must fill out and submit the above survey form on or before the assignment due date.
Course Resources:¶
Written by Dr. Dirk Colbry, Michigan State University
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.