Math 511: Linear Algebra

3.4 Review for Test 1

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3.4.1 Rubrics

Criteria	Excellent	Good	Satisfactory	Needs Improvement
Understanding	Demonstrates a comprehensive understanding of concepts, uses precise terminology, and explains ideas clearly.	Shows a good understanding of concepts, minor inaccuracies in terminology, explanations are generally clear.	Basic understanding of concepts, some errors in terminology, explanations need more clarity.	Limited understanding, frequent errors in terminology, explanations are unclear.
Computational Accuracy	All calculations are correct, logical, and well-organized.	Minor calculation errors, overall approach is sound.	Significant calculation errors affecting results.	Calculations are mostly incorrect or unclear.
Solution Strategy	Selects and applies appropriate strategies effectively.	Solves problems using correct strategies but needs occasional guidance.	Struggles to select or apply appropriate strategies consistently.	Misunderstands or cannot apply concepts to solve problems.
Clear Mathematical Presentation	Presents work in a clear, organized, and logically structured manner, using proper mathematical notation and formatting.	Presentation is generally clear, with minor errors in notation or vocabulary.	Presentation is somewhat unclear or lacks organization, with more frequent errors in notation or vocabulary.	Presentation is difficult to follow, with significant errors in notation or vocabulary.
Linear Algebra	Explicitly connects linear algebra concepts to the solution, highlighting their relevance and demonstrating understanding of their role.	Makes come connections to linear algebra concepts, but could be more explicit or detailed.	Connections to linear algebra concepts are unclear or superficial.	Fails to consistently or accurately connect linear algebra concepts to the solution.
Format	All exam submissions must be in pdf format or they will get a grade of zero

To get any credit for solving a linear system one must

• possibly set up the linear system

• write out the augmented matrix

• reduce the matrix to either

  1. upper triangular form
  2. row echelon form
  3. reduced row echelon form using row operations.
     
     

• use backward substitution

• display the solution as a column vector

  1. angle brackets are used to indicate that the list of numbers is really a column
  2. square or round brackets enclosing a vertical (column) list of numbers and/or variables
  3. square or round brackets enclosing a horizontal list of numbers with $^T$, transpose indicating that the row should be a column vector.
     
     
     

• $(2,1)$ is a point in a two a dimensional space, not a vector in a vector space.

• $x_1=2$, $x_2=1$ is not a vector.

• The correct answer is

$$ \begin{bmatrix} 2 \\ 1 \end{bmatrix} = \left(\begin{matrix} 2 \\ 1 \end{matrix}\right) = \begin{bmatrix} 2 & 1 \end{bmatrix}^T = \left(\begin{matrix} 2 & 1 \end{matrix}\right)^T = \langle 2, 1 \rangle $$

I realize there are other methods of solving these problems because they are small outside of this course. They are only small to allow one to complete the exercise with a reasonable amount of effort.

You must display mastery of the techniques of this course using the methods taught in this course.

I am not looking for a numerically correct answer. I am grading mastery of the techniques taught in this course.

To get any credit computing a determinant

• A $2\times 2$ matrix may be solved using a formula, $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$.

• To compute the determinant of a larger $n\times n$ matrix ($n>2$), one must

  1. Use the ten properties of determinants used in the course
  2. Use the Laplace Expansion, when a row or column has only one nonzero entry

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3.4.2 True/False Questions

To receive all three points for a True/False question one must correctly respond whether the statement is

• True, which means always true

  If the statement is true, then one must additionally explain or prove that answer is true.

• False, otherwise

  If the statement is false, then one must additionally give a numerical answer (no variables) for which the statement is false.

For example

1. For $m\times n$ matrices $A$ and $B$, $A + B = B + A$.

Answer

TRUE

$A + B = [a_{ij}] + [b_{ij}] = [ a_{ij} + b_{ij} ] = [b_{ij} + a_{ij}] = [b_{ij}] + [a_{ij}] = B + A$

2. For $n\times n$ matrices $A$ and $B$, $AB = BA$.

answer

FALSE

Let $A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$, and $B = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$. Then

$$ \begin{align*} AB &= \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \\ \\ BA &= \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \neq AB \end{align*} $$

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3.4.3 What Material is Covered by Test 1

Chapter One

1. Any question from the WebAssign homework, Projects, Problem Sets, or Review Exercises at the end of the chapter in the textbook
2. Any question similar to a question from the WebAssign homework, Projects, Problem Sets, or Review Exercises at the end of the chapter in the textbook

Chapter Two

1. Any question from the WebAssign homework, Projects, Problem Sets, or Review Exercises at the end of the chapter in the textbook
2. Any question similar to a question from the WebAssign homework, Projects, Problem Sets, or Review Exercises at the end of the chapter in the textbook

Chapter Three

1. Any question from the WebAssign homework, Projects, Problem Sets, or Review Exercises at the end of the chapter in the textbook
2. Any question similar to a question from the WebAssign homework, Projects, Problem Sets, or Review Exercises at the end of the chapter in the textbook
3. Any question from the Cumulative Test at the end of chapter 3 in the textbook.

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3.4.4 Test Information

There are two tests:

• a proctored test
• a take-home test.

Proctored Test

• You will have 75 minutes to complete the proctored test.
• There are no notes, calculators, electronic devices, note cards, books, or any aids allowed during the proctored test.
  
  
  

Take-home Test

• The take-home test must be turned on time to receive credit.
• You can use technology to check your work. You must still show all of your work. I am grading the work you complete to compute the answer, not the solution. You must show all of the steps your performed using the techniques we learned in these chapters. "I computed it in my software" is a zero.
• There is no time limit other than the due date and time.
• You must do your own work. A 100% on the take-home test and a 20% on the proctored test averages to a 60% which is still not passing.
  
  
  

Well-prepared means:

• you have your own notes,
• You have memorized the basic formulas, definitions and properties
• You practiced the methods of computing
  1. Gaussian Elimination
  2. Elementary Row Operations and Elementary Matrices
  3. The LU decomposition
  4. The CR decomposition
  5. The solution to a linear system
  6. Determinants
  7. Adjugate
  8. Cramer's Rule
  9. The inverse of a nonsingular matrix
  10. The inverse of a $2\times 2$ nonsingular matrix using the adjugate

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3.4.5 Comments

We will be practicing the skills used in the first two chapters throughout the rest of the course. It is alright to struggle with the first exam. I expect your test scores to get much better and I will only average your Test 1 score into your grade if it raises your grade at the end of the semester.

• I am setting expectations
• You are finding out what the tests look like

The first test is for my information more than yours. You will get better with practice and there will come a time when this all seems as easy as the algebra of scalars.

Unlike your previous Mathematics courses, there are a lot of definitions and concepts to memorize. Give yourself some time. Be patient and keep up the good work.

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Your use of this self-initiated mediated course material is subject to our Creative Commons License 4.0

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