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Applied Linear Algebra for Computer Science and Electrical Engineering

Course Description

In Applied Linear Algebra, you won’t just learn matrices and vectors—you'll experience them in action. This course empowers undergraduate students in Computer Science and Electrical Engineering to develop a deep, functional understanding of linear algebra’s essential role in modern technology, from state-space control systems to machine learning architectures.

Using Generative AI as a dynamic teaching partner, the course transforms learning into a highly personalized, interactive journey. AI tools will craft vivid metaphors, generate immersive stories, create customized content, power simulations, design scaffolded projects, and deliver individualized assessments to optimize every student's growth.

Cognitive Structure (Based on Bloom’s Taxonomy)

1. Remember

  • Knowledge Goals: Recall fundamental concepts such as vector spaces, linear transformations, eigenvalues, and matrix decompositions.
  • AI Enhancement:
  • Generative AI will produce quick-reference analogies, such as comparing a vector basis to different musical scales enabling a song's creation.
  • Interactive AI quizzes will reinforce basic terminology and formulae.

2. Understand

  • Knowledge Goals: Explain how linear algebra models electrical systems, graphics transformations, and data structures.
  • AI Enhancement:
  • AI-generated metaphors (e.g., visualizing range spaces as "pathways" in a robot's movement grid) will make abstract ideas intuitive.
  • AI-led narrative lessons will recount the evolution of linear algebra through the eyes of historical engineers like Claude Shannon and Emmy Noether.

3. Apply

  • Skill Goals: Solve real-world problems by implementing matrix operations, least squares solutions, and eigenvalue analyses.
  • AI Enhancement:
  • AI will create custom practice problems based on each student’s performance, ensuring targeted skill-building.
  • Walkthroughs for matrix manipulations (like LU decomposition) will adapt in complexity based on student mastery.

4. Analyze

  • Skill Goals: Deconstruct systems into matrix models, diagnose system behaviors using eigenstructure, and identify patterns in machine learning datasets.
  • AI Enhancement:
  • Interactive simulations will allow students to model control systems or PCA (Principal Component Analysis) workflows, adjusting inputs and observing outcomes in real-time.
  • AI will support animated breakdowns of complex processes (e.g., illustrating how matrix rank impacts the solvability of a system).

5. Evaluate

  • Skill Goals: Critique different modeling approaches, optimize solutions for stability or efficiency, and assess robustness in electrical systems and ML models.
  • AI Enhancement:
  • AI-driven formative assessments will present alternative problem-solving strategies and prompt students to critique them, encouraging reflective thinking.
  • Personalized feedback will guide students to iterate and improve their work.

6. Create

  • Skill Goals: Design novel applications using linear algebra, such as building a predictive algorithm or engineering a feedback control circuit.
  • AI Enhancement:
  • Project Generation Engine: AI will suggest scaffolded project ideas (e.g., "Design a machine learning model to classify power grid failures") based on student interests and proficiency.
  • Ongoing AI feedback loops will nurture idea refinement and critical problem-solving at each project milestone.

Generative AI: A Key Learning Partner

Metaphors and Analogies

AI will weave fresh metaphors to translate mathematical structures into relatable concepts. Imagine eigenvectors described as "hidden rivers" guiding water (data) through landscapes (systems), enabling intuition to bridge to formal understanding.

Stories and Lessons

Weekly AI-generated episodes will explore the journeys of pioneering figures, framing linear algebra’s historical breakthroughs in captivating, story-driven contexts.

Content Generation

AI will craft: - Real-time quizzes matched to students’ current needs, - Step-by-step walkthroughs of problem solutions, - Dynamic worksheets for matrix computations and system analysis.

Simulations and Animations

Students will: - Manipulate dynamic visual models of matrix transformations. - Experiment with state-space representations in simulated circuits or control systems. - Visualize eigenvalue movements during system perturbations.

Learning Activities and Projects

Students will engage with AI-suggested challenges such as: - Designing circuit feedback systems modeled through state-space equations. - Engineering simple neural network layers built with matrix operations. AI will guide students with scaffolded support and instant feedback that adapts based on performance.

Assessment and Feedback

Going beyond traditional exams, AI will: - Track individual progress against course objectives, - Provide reflective prompts for students to self-assess their learning, - Recommend iterative exercises to build resilience and problem-solving expertise.

Why Take This Course?

By the end of Applied Linear Algebra, you will not only understand the core principles—you will have applied them to build models, analyzed their performance, evaluated different approaches, and created tangible solutions to real-world problems in Computer Science and Electrical Engineering.

With Generative AI as your collaborative guide, you’ll experience a learning journey that’s personalized, engaging, and empowering—preparing you for the challenges of advanced technologies and innovation-driven careers.