# MATH310S26 - Introduction to Mathematical Modeling (Lecture Notes) ## Spring 2026 - Day 1 Notes ### Date: January 12, 2026 (01/12/26) --- ## Course Information ### Instructor - **Name**: Scott Strong - **Email**: [email protected] - **Office**: Stratton Hall 205 - **Office Hours**: - Mondays: 3:00-5:00 PM (Drop-in, Stratton 102 or 205) - Tuesdays: 1:00-3:45 PM (Drop-in, Stratton 102 or 205) - Wednesdays: 3:00-5:00 PM (By appointment - tinyurl.com/sstrongOHS26) ### Course Details - **Course**: MATH 310 - Introduction to [mathematical modeling](https://en.wikipedia.org/wiki/Mathematical_model) - **Term**: Spring 2026 - **Meeting Time**: MWF 1:00-1:50 PM - **Location**: Berthoud 205 - **Final Presentations**: May 8th, 10:15 AM - 12:15 PM (for final presentations) --- ## Course Description & Objectives ### Overview - Introduction to [mathematical modeling](https://en.wikipedia.org/wiki/Mathematical_model) and communication in mathematics - Writing-intensive course (symbols and words) - Transition from core math sequence to upper division [applied mathematics](https://en.wikipedia.org/wiki/Applied_mathematics) and [applied statistics](https://en.wikipedia.org/wiki/List_of_fields_of_application_of_statistics) curriculum - Focus on formulating, solving, and presenting applied problems ### Learning Outcomes 1. Formulate and investigate mathematical and [statistical models](https://en.wikipedia.org/wiki/Statistical_model) 2. Identify multiple types of models and techniques 3. Communicate results in writing and orally 4. Integrate software tools with mathematical thinking --- ## Grading Structure ### Grade Requirements #### Pass (D Level) - Attend 75% of all classes (sans excused absences) (LA) - Complete 75% of Lecture Reflections (LR) - Complete 75% of Low-stakes Assessments/Feedback (LSF) #### C Level - Everything from D level - Participate in 90% of interactive work days (every third day) (IW) - Complete additional activities (readings, etc.) (AA) #### B Level - Everything from C level - Report and presentation about an existing mathematical model - Can be individual or group work #### A Level - Everything from B level - Create mathematical modeling project from scratch OR - Advance a part of an existing mathematical model --- ## Important Due Dates ### Wednesday, January 14, 2026 1. **Student Bio Slide** - Create a slide introducing yourself 2. **Sticker Prompt** - Submit AI prompt for personalized attendance stickers (36 stickers) 3. **Lecture Reflection (LR) Discussion** - Discuss the reflection process --- ## Lecture Content: [Linear algebra](https://en.wikipedia.org/wiki/Linear_algebra) and [systems of linear equations](https://en.wikipedia.org/wiki/System_of_linear_equations) ### Matrix-Vector Multiplication #### Setup Given [matrix](https://en.wikipedia.org/wiki/Matrix_(mathematics)) $A \in \mathbb{R}^{2 \times 2}$ and [vector](https://en.wikipedia.org/wiki/Vector_(mathematics_and_physics)) $\mathbf{x} \in \mathbb{R}^{2}$ (with entries in the [real numbers](https://en.wikipedia.org/wiki/Real_number)): $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}$ where $a, b, c, d, x, y \in \mathbb{R}$ #### [Matrix multiplication](https://en.wikipedia.org/wiki/Matrix_multiplication) The product $A\mathbf{x}$ is computed as: $A\mathbf{x} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}$ #### [Linear combination](https://en.wikipedia.org/wiki/Linear_combination) Interpretation This can be rewritten as a linear combination of columns: $A\mathbf{x} = x\begin{bmatrix} a \\ c \end{bmatrix} + y\begin{bmatrix} b \\ d \end{bmatrix} = x \cdot \mathbf{a}_1 + y \cdot \mathbf{a}_2$ Where: - $\mathbf{a}_1 = \begin{bmatrix} a \\ c \end{bmatrix}$ is the first column of $A$ - $\mathbf{a}_2 = \begin{bmatrix} b \\ d \end{bmatrix}$ is the second column of $A$ ### Homogeneous Linear Systems #### Setting $A\mathbf{x} = \mathbf{0}$ When we set $A\mathbf{x} = \mathbf{0}$, we get the system: $\begin{align} ax + by &= 0 \\ cx + dy &= 0 \end{align}$ #### Solving for y From these equations: - From equation 1: $y_1(x) = -\frac{a}{b}x$ - From equation 2: $y_2(x) = -\frac{c}{d}x$ Both lines pass through the origin $(0,0)$ with y-intercept $= 0$. ### Non-Trivial Solutions #### Key Question Can there be more than just the $(0,0)$ solution? Answer: Yes, when the two lines have the same [slope](https://en.wikipedia.org/wiki/Slope)! #### Condition for Same Slope For the lines to coincide (have infinitely many solutions): - Slopes must be equal: $-\frac{a}{b} = -\frac{c}{d}$ - Cross-multiplying: $-ad = -bc$ - Rearranging: $ad - bc = 0$ #### The [determinant](https://en.wikipedia.org/wiki/Determinant) The expression $\det(A) = ad - bc$ is the determinant of matrix $A$. Key Result: The system $A\mathbf{x} = \mathbf{0}$ has non-trivial solutions when $\det(A) = 0$ (i.e., $A$ is a [singular matrix](https://en.wikipedia.org/wiki/Singular_matrix)). ### Geometric Interpretation When $\det(A) = 0$: 1. The two lines are the same line ([collinear](https://en.wikipedia.org/wiki/Collinearity)) 2. The column vectors $\mathbf{a}_1$ and $\mathbf{a}_2$ point in the same direction 3. The columns are [linearly dependent](https://en.wikipedia.org/wiki/Linear_independence#Linear_dependence) 4. There exists $\lambda \in \mathbb{R}$ such that $\mathbf{a}_2 = \lambda \mathbf{a}_1$ (the [column space](https://en.wikipedia.org/wiki/Column_space) is one-dimensional) ### Check Your Understanding Problem: Show that if $\det(A) = ad - bc = 0$, then the column vectors $\mathbf{a}_1 = \begin{bmatrix} a \\ c \end{bmatrix}$ and $\mathbf{a}_2 = \begin{bmatrix} b \\ d \end{bmatrix}$ are collinear (point in the same direction) and are therefore [linearly dependent](https://en.wikipedia.org/wiki/Linear_independence#Linear_dependence). Note: In differential equations, this relates to the [Wronskian](https://en.wikipedia.org/wiki/Wronskian) — when vectors/functions point in the same direction, they are not [linearly independent](https://en.wikipedia.org/wiki/Linear_independence). --- ## Course Structure Notes ### Attendance - Based on collecting personalized stickers (fun, not lame) - 36 individualized stickers throughout semester - Generated from student-submitted AI prompts ### Interactive Work Days - Every third day of class - Break from lecture content - Work on mathematical modeling problems - Collaborate with peers ### Projects - Choose topics based on personal interest - Natural creativity often leads to "going too big" - Part of learning process to scope down to manageable sub-problems - Focus on communication of findings (symbols and context) ### Assessment Philosophy - Low-stakes continuous assessment - Focus on feedback loops - Presentations ensure human verification in AI era - Autonomy and choice while maintaining structure --- ## Next Steps 1. Submit Student Bio Slide by Wednesday 2. Submit Sticker Prompt for attendance tracking 3. Prepare for Lecture Reflection discussion 4. Review linear algebra concepts: - [Matrix multiplication](https://en.wikipedia.org/wiki/Matrix_multiplication) - [Determinants](https://en.wikipedia.org/wiki/Determinant) - [Linear independence](https://en.wikipedia.org/wiki/Linear_independence) - [Systems of linear equations](https://en.wikipedia.org/wiki/System_of_linear_equations) and geometric interpretation