---
tags: mth302
---
# Skill Quiz 2
This quiz contains the **second version** of **Foundational Skill LA.1** and the **first version** of **Skill LA.2**.
**Instructions:**
* If you had a "Success" mark on Skill LA.1 from the first quiz, **do not do the problem for that skill again**.
* Make sure to consult the [Standards for Student Work in MTH 302](https://github.com/RobertTalbert/linalg-diffeq/blob/main/course-docs/standards-for-student-work.md) document before starting on your work, so you're clear on what is expected and what constitutes a "successful" attempt. Also check the *Success criteria* below each problem.
* This week's quiz is done entirely asynchronously since we are not meeting as a class. Please submit your work on Blackboard by **11:59pm ET Monday, January 30**.
* As before, you may hand-write your work on paper, hand-write it in a notes app, or type it up. But **for this and all subsequent quizzes, put the work for each Foundational Skill on its own page.** So, if you are doing both LA.1 and LA.2, put the work for LA.1 on a different page than the work for LA.2. *Do not put both problems on the same page.*
* When you are ready to submit your work: **Scan** your handwritten work to a clear, legible, black-and-white PDF using a scanner or scanning app -- **one PDF per problem**. So if you are doing both problems, you will have two PDFs: one for Skill LA.1 and another for Skill LA.2 (all parts). Then, **upload each PDF to its designated folder** on Blackboard: The PDF for Skill LA.1 goes into the folder for Skill LA.1, and the PDF for Skill LA.2 goes into the folder for Skill LA.2. Make sure to click "Submit" on each before exiting.
* Your work will be graded on Blackboard, receiving a mark of either *Success* or *Revise* along with feedback. If you need to *Revise*, a new version of each of these skills will appear on Skill Quiz 3.
---
## Foundational Skill LA.1
>LA.1: I can solve a system of linear equations by converting it into an augmented matrix and putting into reduced row echelon form.
Consider the system:
$$\begin{alignedat}{4}
x & {}+{} & y & {}+{} & 2z & {}={} & 1 \\
2x & {}-{} & y & {}-{} & 2z & {}={} & 2 \\
-x & {}+{} & y & {}+{} & z & {}={} & 0 \\
\end{alignedat}$$
1. Convert this system into an augmented matrix.
2. Then, using a sequence of row operations, transform the matrix into reduced-row echelon form (RREF).
3. Finally, determine whether the linear system has exactly one solution, infinitely many solutions, or no solutions. If there is one solution, state what it is. If there are infinitely many solutions, state the free variable(s) in the system and then express the pivots in terms of the free variables. If there are no solutions, explain how you know from the RREF of the matrix.
*Success criteria:* The augmented matrix is correct with no copy or typographical errors. Each row operation in getting the matrix to RREF is clearly indicated, and no more than two simple errors are present. The number of solutions is correct given the RREF; and if the system is consistent, the solutions are correctly and clearly stated. **Note**: No other solution methods, such as substitution or graphical solutions, are allowed.
---
## Foundational Skill LA.2
> LA.2: I can determine if a vector is in the span of a collection of other vectors.
Below are three sets of vectors along with an additional target vector $\mathbf{b}$. For each, determine whether $\mathbf{b}$ belongs to the span of the given vectors. Do this using matrices and row-reduction. *You may do all row-reduction steps with a computer*, but in your writeup, you must clearly state the matrix you are row-reducing, the resulting matrix after row-reduction, and a single sentence that answers the question (Is $\mathbf{b}$ in the span of the given vectors?) and explains how you know based on the computation.
1. Given vectors: $\mathbf{v_1} = [1 \ 2]^T$, $\mathbf{v_2} = [-6 \ -12]^T$; target vector $\mathbf{b} = [5 \ 5]^T$
2. Given vectors: $\mathbf{v_1} = [1 \ 2 \ 5]^T$, $\mathbf{v_2} = [-3 \ {-6} \ 4 ]^T$, $\mathbf{v_3} = [0 \ 2 \ {-1}]^T$; target vector $\mathbf{b} = [5 \ 10 \ 15]^T$
3. Given vectors: $\mathbf{v_1} = [1 \ 2 \ 5 \ -1]^T$, $\mathbf{v_2} = [{-3} \ {-6} \ 4 \ 2]^T$, $\mathbf{v_3} = [1 \ 0 \ {-1} \ 0]^T$; target vector $\mathbf{b} = [{-1} \ {-4} \ 8 \ 1]^T$
*Success criteria:* Matrix methods have been used appropriately to answer the questions. The matrix being used and the results of row-reduction are clearly stated. The answer (yes/no) to each question is clearly stated and is consistent with the results of the matrix operations. Each item has not only an answer but also a clear one-sentence explanation in English that explains the answer correctly in terms of the matrix and row-reduction.

:::info Welcome to MTH 201! I'm Dr. Robert Talbert, Professor of Mathematics, and I am grateful that you are signed up for the course and am looking forward to working with you this semester. ::: What's MTH 201 all about? MTH 201 is a first course in Calculus, which is all about modeling and understanding change. Change is maybe the most important facet of the world around us, and we care about it more than we realize. For example, we care a lot about the number of Covid-19 cases in our community, but we might care even more about how fast the number of cases is changing (either up or down). In MTH 201, you'll learn the mathematical language of change and apply it to models that you build to draw conclusions, make predictions, and give meaningful answers to real problems. MTH 201 goes beyond just computation. In MTH 201, you'll build skills with understanding complex concepts, communicating those concepts and the meaning of your results to appropriate audiences, using professional tools to help you in your work, and practice working with others to improve your learning (and theirs). These are valuable skills no matter where you go next. Success in this course doesn't come easy, and you can expect to be pushed and stretched intellectually. But the struggle you experience is normal and healthy, a sign of growth and that you are doing things the right way. And you will receive tireless support from me and your classmates in the process. Above all, my top priority is to support you in your work and help you succeed.

11/11/2023Initial due date: Sunday, April 9 at 11:59pm ET Overview Our final miniproject reaches back into linear algebra to look at diagonalizable matrices and their uses in solving systems of differential equations. Prerequisites: You'll need to be able to solve basic systems of differential equations and find the eigenvalues and eigenvectors for a small matrix. You'll also need a basic comfort level with concepts of linear independence and matrix arithmetic from earlier in the course. Background This entire problem comes from Section 3.9.1 in your textbook. Here is a rephrased version of the introduction to that section.

3/29/2023Initial due date: Sunday, April 9 at 11:59pm ET Overview This miniproject will teach you about the Runge-Kutta method, a standard numerical solution technique for differential equations. Prerequisites: A strong grasp of Euler's Method for single DE's is needed. You will also need to be comfortable using a spreadsheet. Miniproject 6 (Euler's Method for systems) is also recommended. Background A description of the Runge-Kutta method along with an example is given in this tutorial. Read it carefully and make sure you can work along with the example before proceeding.

3/29/2023Initial due date: Sunday, April 9 at 11:59pm ET Overview This miniproject introduces a version of Euler's Method as a numerical solution technique for systems. Prerequisites: You will need to be comfortable with using Euler's method for single differential equations. You'll also benefit from some familiarity with spreadsheets or Python in order to automate the calculations. Background This tutorial gives you the background you need for this assignment. Please read it and make sure you understand the concepts and the example: https://github.com/RobertTalbert/linalg-diffeq/blob/main/assignments/Euler's_Method_for_Systems.ipynb

3/22/2023
Published on ** HackMD**

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