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: Fundamental Spaces¶
Image from: https://kevinbinz.com/2017/02/20/linear-algebra/
Goals for today's pre-class assignment¶
1. Ideas for reducing stress¶
The university, faculty and your instructors have been listening to feedback from students about the difficulties and stress related to on-line learning. We are concerned about student health and acknowledge that this semester is much more difficult than most. We would like to work with you to try and find additional ways we can help you reduce stress and be successful this semester. As a starting point, here is what we have tried to do so far:
When designing this course to be on-line this semester, the instructors implemented the following changes:
- Instead of two high-point, stressful, midterm exams we moved to a low-point bi-weekly quiz model. These quizzes are intended to be lower stress since they are each only worth about 2% of your final grade and we plan to automatically drop the lowest quiz (quizzes are 10% of your final grade, 6 quizzes and 1 drop, so each quiz is only 2% of your final grade).
- We aligned curriculum in all three sections of the course to sync up assignments and enable instructors to help out students in other sections.
- With 10 instructors, we have added a significant amount of additional office hours available during the week. Instructors are also encouraged to answer questions over slack and email.
- We implemented a regular homework/quiz routine to help students schedule their time.
- We used the quiz days to try and help slow down the schedule and space out the topics.
- We integrated 3 open textbooks into the curriculum to give students plenty of resources for individual learning and practice. Having three textbooks also provides variety to help match students' different learning styles.
During the semester we also made some additional course adjustments based on our observations and feedback from students (pre-class surveys, emails, office hours, and anonymous surveys). These changes include the following:
- Adjusting in-class assignments so we can lecture less and encourage more hands-on support during class.
- Moved away from random groups and started using assigned groups (intended to help students build relationships).
- Updated the format for the most recent homework (HW4) to include more practice using the quiz-style questions.
Even given the above, we feel there is still room for improvement and would like your input. During the next class we plan to spend some time to brainstorm ideas from you about things that we can do to make the class better. Ideally, we are looking for actionable ideas we can try to implement now that will help reduce stress for all students, while still ensuring that we meet the class learning goals.
Please note, there is rarely a "one-size-fits-all" solution and we need to ensure that any changes we implement support most or all students and not just a select few. We also need to be aware that any change we make may have unintended consequences. For example:
- We can't easily just "drop" additional assignments. Every time we drop an assignment, the point value for existing assignments goes up. This will effectively increase the stress for existing assignments.
- We still need to be able to evaluate your understanding of the material. We need this evaluation to a) provide more time / feedback to students on topics that are being missed, 2) ensure we can prepare you for future classes that require mastery of the topics and 3) provide value for your degree and future employment.
✅ Do This: Spend some time to think about ways that the MTH314 instructors can help students reduce their stress while maintaining the learning goals for the class. Write these ideas down in the following cell and be prepared to share them with your instructors in class. If you are uncomfortable sharing as a group, please always feel free to reach out to the course coordinator (Dirk Colbry colbrydi@msu.edu) with your suggestions.
Put your ideas and notes here.
2. Orthogonal Complement¶
Definition: A vector $u$ is orthogonal to a subspace $W$ of $R^n$ if $u$ is orthogonal to any $w$ in $W$ ($u\cdot w=0$ for all $w\in W$).
For example, consider the following figure, if we consider the plane to be a subspace then the perpendicular vector comming out of the plane is is orthoginal to any vector in the plane:
Definition: The orthogonal complement of $W$ is the set of all vectors that are orthogonal to $W$. The set is denoted as $W_{\bot}$.
✅ **QUESTION:** Is $W_\bot$ a subspace of $R^n$? Justify your answer briefly.
Put your answer to the above question here
✅ **QUESTION:** What are the vectors in both $W$ and $W_\bot$?
Put your answer to the above question here
from IPython.display import YouTubeVideo
YouTubeVideo("5B8XluiqdHM",width=640,height=360, cc_load_policy=True)
Projection of a Vector onto a Subspace¶
Think of a projection onto a subspace is analogous to a shadow on a surface. Aspects of an objects 3D space is represented in a 2D shadow but you can't take the shadow by itself and exactly recreate the 3D surface.
Image from https://commons.wikimedia.org
The following is the matimatical defination of projection onto a subspace.
Definition: Let $W$ be a subspace of $R^n$ of dimension $m$. Let $\{w_1,\cdots,w_m\}$ be an orthonormal basis for $W$. Then the projection of vector $v$ in $R^n$ onto $W$ is denoted as $\mbox{proj}_Wv$ and is defined as $$\mbox{proj}_Wv = (v\cdot u_1)u_1+(v\cdot u_2)u_2+\cdots+(v\cdot u_m)u_m$$
Another way to say the above defination is that the project of $v$ onto the $W$ is just the sumation of $v$ projected onto each vector in a basis of $W$
Remarks:
Recall in the lecture on Projections, we discussed the projection onto a vector, which is the case for $m=1$. We used the projection for $m>1$ in the Gram-Schmidt algorithm.
The projection does not depend on which orthonormal basis you choose.
If $v$ is in $W$, we have $\mbox{proj}_Wv=v$.
The Orthogonal Decomposition Theorem¶
Theorem: Let $W$ be a subspace of $R^n$. Every vector $v$ in $R^n$ can be written uniquely in the form $$v= w+w_{\bot},$$ where $w$ is in $W$ and $w_\bot$ is orthogonal to $W$ (i.e., $w_\bot$ is in $W_\bot$). In addition, $w=\mbox{proj}_Wv$, and $w_\bot = v-\mbox{proj}_Wv$.
Definition: Let $x$ be a point in $R^n$, $W$ be a subspace of $R^n$. The distance from $x$ to $W$ is defined to be the minimum of the distances from $x$ to any point $y$ in $W$. $$d(x,W)=\min \{\|x-y\|: \mbox{ for all }y \mbox{ in } W\}.$$ The optimal $y$ can be achieved at $\mbox{proj}_Wx$, and $d(x,W)=\|x-\mbox{proj}_Wx\|$.
✅ **QUESTION:** Let $v=(3, 2, 6)$ and $W$ is the subspace consisting all vectors with the form $(a, b, b)$. Find the projection of $v$ onto $W$.
Put your answer to the above question here
✅ **QUESTION:** Let $v=(3, 2, 6)$ and $W$ is the subspace consisting all vectors with the form $(a, b, b)$. Find the distance from $v$ to $W$.
Put your answer to the above question here
3. The Four Fundamental Spaces¶
In the lecture on Change Basis, we talked about four subspaces based on a matrix $A$:
Row space of $A$: linear combination of all rows of $A$
Column space of $A$: linear combination of all columns of $A$
Null space or kernel of $A$: all $x$ such that $Ax=0$
Null space of $A^\top$: all $y$ such that $A^\top y =0$
In this course we represent a system of linear equations as $Ax=b$. The matrix $A$ can be viewed as taking a point $x$ in the input space and projecting that point to $b$ in the output space.
It turns out, everything we need to know about $A$ is represented by four fundamental vector spaces. Two of the four spaces are easily defined as follows:
Row space of $A$: linear combination of all rows of $A$
Column space of $A$: linear combination of all columns of $A$
The other two fundamental spaces are defined by a concept called the Null Space. The Null space is calculated by finding all the solutions to the homogeneous system $Ax=0$. The final two fundamental spaces are defined as follows:
Null space or kernel of $A$: all $x$ such that $Ax=0$
Null space of $A^\top$: all $y$ such that $A^\top y =0$
4. Independent Learning¶
✅ **DO THIS:** Find a YouTube video that helps you understand the four fundamental spaces.
✅ **QUESTION:** What is the URL for your video?
Put your answer to the above question here
✅ **DO THIS:** Add the link to the video to the code below. Try embedding the link in the provided Python YouTubeVideo Function by replacing XXXXX with the video ID.
from IPython.display import YouTubeVideo
YouTubeVideo("XXXXXX",width=640,height=360, cc_load_policy=1)
✅ **QUESTION:** What criteria did you use in selecting your video?
Put your answer to the above question here
✅ **QUESTION:** How long into a video did you go before deciding if it was good or bad?
Put your answer to the above question here
✅ **QUESTION:** What did you like about the video you selected.
Put your answer to the above question here
✅ **QUESTION:** What didn't you like about the video?
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:** What is the URL for your video for the four Fundamental spaces?
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
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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 Dirk Colbry, Michigan State University
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.