--- 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.
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Robert Talbert Professor of Mathematics, Grand Valley State University. Writer (http://rtalbert.org, http://rtalbert.org/book), speaker, researcher.
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Miniproject 6

Initial 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
Miniproject 4

Initial due date: Sunday, March 26 at 11:59pm ET Overview Eigenvalues of a matrix are incredibly useful and important for many applications. (Some of these applications are in Miniprojects 1-3.) But computing eigenvalues of a matrix, even of relatively small size, can be difficult or impossible to do exactly. So we need numerical approximation methods for most practical uses of eigenvalues. This miniproject will teach you one such method. Prerequisites: You'll need to know what an eigenvalue and eigenvector for a matrix are, and how to find these using SymPy. You'll also need to know how to multiply matrices and vectors. Background Complete the following warmup exercises first. These don't go in your writeup. They are just here to teach you some terminology you'll need in the main assignment.

3/3/2023
Miniproject 3

Initial due date: Sunday, March 12 at 11:59pm ET Overview In this miniproject, you'll apply concepts from linear algebra to study discrete dynamical systems. These are related to the idea of Markov chains that was the subject of Miniproject 1 and are the linear algebra analogue of systems of differential equations which we will study later. Prerequisites: You'll need to know what an eigenvalue and eigenvector for a matrix are, and how to find these using SymPy. This miniproject also requires some knowledge of matrix-vector multiplication. Background Complete the following before beginning this miniproject. These are not part of your writeup, but you'll need the knowledge before you can understand the tasks in the assignment.

2/21/2023
Miniproject 2

Initial due date: Sunday, February 26 at 11:59pm ET Overview In this miniproject, you'll explore the concept of matrix transformations and use matrix transformations in $\mathbb{R}^3$ to create a very simple computer animation --- and see how linear algebra is used to make more complex animations happen. Prerequisites: You'll need to know how to multiply a $3 \times 3$ matrix to a $3 \times 1$ vector using SymPy, as well as what some of our recent theorems about matrices and invertibility say. Also, you'll need to do a bit of background learning using the video mentioned below. Background Complete the following before beginning this miniproject. These are not part of your writeup, but you'll need the knowledge before you can understand the tasks in the assignment.

2/9/2023
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