Module 4: Linear Algebra: Case Studies
These cases show vectors and matrices as tools for representation, transformation, and measurement.
Case Study 1: User Preferences as Vectors
Scenario: A recommendation prototype stores user interests as free-form tags. Similar users are hard to compare because every profile has a different shape.
Source anchor: Khan Academy: Matrix transformations.
Module concepts:
- vectors
- dimensions
- dot product intuition
- normalization
Wrong Approach
Compare profiles by reading tags manually.
Better Approach
Represent each user as a vector over a shared feature space. Use dot product or cosine similarity with normalization so high-activity users do not dominate only because they have more entries.
Tradeoff Table
| Choice | Gain | Cost |
|---|---|---|
| Free-form tags | Flexible | Hard to compare |
| Feature vectors | Computable similarity | Needs feature design |
| Learned embeddings | Richer semantics | Harder to inspect |
Failure Mode
The system recommends users with many tags rather than similar interests.
Required Artifact
Create a feature vector table for three users and compute one similarity comparison by hand.
Project / Capstone Connection
Use this vector-table format later when explaining recommendation, ranking, embedding, or feature-space choices in capstone artifacts.
Case Study 2: Image Transformations With Matrices
Scenario: A simple graphics tool rotates and scales icons. A learner applies transformations in the wrong order and gets unexpected placement.
Source anchor: Khan Academy: Matrix transformations.
Module concepts:
- matrix transformations
- composition order
- coordinate systems
- geometric interpretation
Wrong Approach
Treat rotation and scaling as interchangeable operations.
Better Approach
Represent each transformation as a matrix and multiply in the correct order for the coordinate convention. Test with a simple point so the visual result is explainable.
Tradeoff Table
| Choice | Gain | Cost |
|---|---|---|
| Trial-and-error transforms | Quick visual feedback | No reliable reasoning |
| Matrix model | Predictable composition | Requires convention clarity |
| Graphics library only | Less math code | Can hide order mistakes |
Failure Mode
An icon scales around the wrong origin after a rotation.
Required Artifact
Write a two-step transformation calculation for one point and explain why reversing the order changes the result.
Project / Capstone Connection
Carry this transformation-order explanation into graphics, simulation, or data-pipeline work where composition order changes meaning.
Case Study 3: Solving Linear Constraints in Resource Planning
Scenario: A small team allocates hours across product, support, and maintenance. Constraints are linear, but the plan is adjusted by intuition and repeatedly overcommits people.
Source anchor: Khan Academy: Matrix transformations.
Module concepts:
- systems of equations
- matrices
- constraints
- feasibility
Wrong Approach
Change one allocation at a time without checking the full system.
Better Approach
Model the constraints as equations or inequalities, solve the system when possible, and identify when the requested work is infeasible.
Tradeoff Table
| Choice | Gain | Cost |
|---|---|---|
| Manual adjustment | Fast discussion | Hidden infeasibility |
| Linear model | Clear constraints | Simplifies reality |
| Optimization tool | Handles more cases | More setup |
Failure Mode
The plan looks balanced in one category but exceeds total available hours.
Required Artifact
Create a matrix or equation system for three work categories and solve or mark infeasible.
Project / Capstone Connection
Reuse this feasibility framing later when project scope, staffing, or resource allocation problems need a defensible constraint model.
Case Study 4: Movement In A 2D Game Uses Vector Addition
Scenario: A simple game moves a character by updating x and y separately in ad hoc ways. Diagonal movement is faster than horizontal movement, and knockback effects stack strangely.
Source anchor: 3Blue1Brown: Essence of Linear Algebra is a useful intuition-first anchor for vectors as direction-and-magnitude objects rather than just number pairs.
Module concepts:
- vector addition
- magnitude
- normalization
- coordinate representation
Wrong Approach
Treat each axis update as an unrelated special case.
Better Approach
Represent movement and force as vectors, add them, then normalize where the game logic requires constant-speed movement independent of direction.
Tradeoff Table
| Choice | Gain | Cost |
|---|---|---|
| separate axis tweaks | quick to prototype | inconsistent behavior |
| vector model | coherent movement math | requires geometric thinking |
| normalized movement | stable speed | extra computation and conventions |
Failure Mode
Diagonal movement becomes unintentionally stronger because the combined vector magnitude is larger than a single-axis move.
Required Artifact
Compute the resulting vector for right movement plus upward knockback, then compare raw and normalized magnitudes.
Project / Capstone Connection
Use this vector-addition explanation later in graphics, simulation, robotics, or recommendation work where combining signals has real geometric meaning.
Case Study 5: Feature Scaling Distorts Similarity
Scenario: A matching tool compares candidates using years of experience, interview score, and portfolio count. Experience ranges from 0 to 20 while the other features are much smaller, so one dimension dominates the similarity calculation.
Source anchor: 3Blue1Brown: Essence of Linear Algebra gives the right intuition for vectors as geometric objects whose direction and magnitude both matter.
Module concepts:
- vector magnitude
- scaling
- similarity
- normalization
Wrong Approach
Compute similarity directly on raw feature values without inspecting scale differences.
Better Approach
Choose a shared feature space, normalize or rescale dimensions where needed, and explain whether magnitude should represent strength, volume, or merely measurement units.
Tradeoff Table
| Choice | Gain | Cost |
|---|---|---|
| raw feature vectors | minimal preprocessing | one dimension can dominate |
| normalized vectors | fairer comparison | loses some magnitude information |
| weighted scaling | domain-aware control | requires justified weights |
Failure Mode
The system claims two candidates are similar mainly because one large-scale feature overwhelms the rest of the vector.
Required Artifact
Construct two candidate vectors, compute similarity before and after scaling, and explain what changed.
Project / Capstone Connection
Use this scaling explanation later in search, recommendation, ranking, or ML-feature discussions where numeric representation choices shape system behavior.
Source Map
| Source | Use it for |
|---|---|
| Khan Academy: Matrix transformations | Anchoring transformation order, matrix composition, and geometric reasoning in approachable examples. |
| 3Blue1Brown: Essence of Linear Algebra | Building intuition for vectors, scaling, magnitude, and linear transformations as geometric objects rather than disconnected formulas. |