Math Lab (Linear Algebra + Probability) — FAANG-Level Problem Set

Goal: Consolidate Week 2 with derivations + short computational checks.

Outcome: Students can derive key results, verify with small code experiments, and communicate clearly under interview constraints.

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How to Start

1. Fork this repository.
2. Open math_student_lab.ipynb in Google Colab.
3. Complete derivations + TODO code.
4. Restart runtime → Run All cells.
5. Push changes and submit a Pull Request.

⚠️ Do NOT edit notebooks directly on GitHub.

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Lab Rules (FAANG Style)

• ✅ Show steps (but be concise)
• ✅ Validate with a small numeric experiment
• ✅ Call out assumptions

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Notebook Rules

• Do NOT rename the notebook
• Do NOT delete TODOs
• Do NOT hardcode outputs
• Notebook must run top-to-bottom

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Dataset

• Synthetic matrices/vectors and toy probability setups

Why?

• Keeps focus on derivation + verification
• Mirrors interview whiteboard + quick sanity check workflow

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Section 1 — Projection Matrices

Task 1.1: Prove/verify projection properties

Checkpoint Questions:

• What does idempotent mean?
• Why is orthogonal projection symmetric?

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Section 2 — Positive Semidefinite (PSD) Matrices

Task 2.1: Show X^T X is PSD

Interview Angle:

• Where does PSD show up in ML (covariance, kernels)?

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Section 3 — Least Squares

Task 3.1: Derive normal equations + verify numerically

FAANG Gotcha:

• Avoid matrix inverse; use solve/lstsq.

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Section 4 — Bayes + Base Rate

Task 4.1: Derive P(D|+) + simulate

Checkpoint Questions:

• Why does base rate dominate when prevalence is low?

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Section 5 — PCA Link

Task 5.1: Covariance eigenvectors vs SVD directions

Interview Angle:

• How does PCA relate to embeddings and compression?

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Submission Expectations

Students must submit:

• Derivations written clearly
• TODO code complete
• Checks passing

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FAANG Interview Evaluation Rubric

Skill	Evaluated
Derivation correctness	✅
Verification discipline	✅
ML intuition	✅
Explanation clarity	✅

Topics

• Linear algebra: projections, orthogonality, PSD matrices
• Probability: expectation, variance, Bayes
• ML links: least squares, PCA, Naive Bayes