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📢 ANNOUNCEMENTS¶

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TEXTBOOK AND REFERENCES¶

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COURSE MATERIALS¶

• 📄 LECTURES - Lecture slides and notes in PDF format
• 📐 MATH TUTORIALS - Detailed mathematical derivations and notes
• 💻 NOTEBOOKS - Python/Jupyter implementations and exercises

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COURSE STRUCTURE & ASSESSMENT¶

6 Core Chapters (Focus)¶

#	Topic	Description
1	Linear Algebra	Vectors, matrices, and operations
2	Analytic Geometry	Geometric interpretations
3	Matrix Decomposition	Eigendecomposition & SVD
4	Vector Calculus	Gradients & optimization
5	Probability & Distributions	Statistical foundations
6	Optimization	Model training & parameter estimation

Assessment Breakdown¶

Component	Weight	Details
Quizzes	40%	4-5 quizzes throughout course
Midterm Examination	20%	-
Final Examination	30%	-
Reading & Review Papers	10%	4-5 papers
TOTAL	100%	-

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EXAM QUESTION PHILOSOPHY & EXAMPLES¶

Sample Exam Question¶

This question exemplifies the depth and rigor expected in exams, requiring synthesis of multiple concepts and progressive problem-solving.

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Linear Independence, Basis, and Rank¶

Consider the matrix \(\mathbf{A} \in \mathbb{R}^{4 \times 4}\) with column vectors: $\(\mathbf{A} = \begin{bmatrix} 1 & 2 & 3 & 5 \\ 2 & 1 & 3 & 4 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 2 \end{bmatrix}\)$

(a) Use Gaussian elimination to determine \(\text{rank}(\mathbf{A})\) by reducing to row echelon form. Identify which columns form a basis for the column space \(\mathcal{C}(\mathbf{A})\) and express each remaining column as a linear combination of these basis columns.

(b) Find a basis for the null space \(\mathcal{N}(\mathbf{A}) = \{\mathbf{x} \in \mathbb{R}^4 : \mathbf{Ax} = \mathbf{0}\}\) by solving the homogeneous system. Verify that \(\text{rank}(\mathbf{A}) + \dim(\mathcal{N}(\mathbf{A})) = 4\).

(c) Prove the following general statement: If \(\mathbf{A} \in \mathbb{R}^{m \times n}\) has rank \(r\), then any set of \(r+1\) columns must be linearly dependent. Apply this to show that columns 1, 2, and 4 of your matrix cannot all be part of a linearly independent set if \(r < 3\).

(d) Compute the row space \(\mathcal{R}(\mathbf{A})\) by finding a basis for the row space from the row echelon form. Show that \(\dim(\mathcal{R}(\mathbf{A})) = \dim(\mathcal{C}(\mathbf{A}))\).

(e) Demonstrate that \(\mathcal{N}(\mathbf{A})\) and \(\mathcal{R}(\mathbf{A})\) are orthogonal subspaces by verifying that every vector in \(\mathcal{N}(\mathbf{A})\) is orthogonal to every row of \(\mathbf{A}\). What does this tell you about the decomposition \(\mathbb{R}^4 = \mathcal{R}(\mathbf{A}^T) \oplus \mathcal{N}(\mathbf{A})\)?

(f) Consider the augmented matrix \([\mathbf{A} | \mathbf{b}]\) where \(\mathbf{b} = [1, 1, 0, 1]^T\). Without solving the system, use your knowledge of \(\mathcal{C}(\mathbf{A})\) to determine whether \(\mathbf{b} \in \mathcal{C}(\mathbf{A})\). If \(\mathbf{b} \notin \mathcal{C}(\mathbf{A})\), decompose \(\mathbf{b}\) as \(\mathbf{b} = \mathbf{b}_{\parallel} + \mathbf{b}_{\perp}\) where \(\mathbf{b}_{\parallel} \in \mathcal{C}(\mathbf{A})\) and \(\mathbf{b}_{\perp} \perp \mathcal{C}(\mathbf{A})\).

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LECTURE STRUCTURE: THREE-PART APPROACH¶

PART 1: Concepts & Explanation¶

Theoretical foundations and intuitive understanding of the topic. - Key definitions - Conceptual framework - Intuitive explanations - Real-world context

PART 2: Mathematical Examples & Tutorials¶

Hands-on mathematical work with students through guided examples. - Worked examples - Step-by-step solutions - Interactive tutorials - Mathematical derivations

PART 3: Python Implementation¶

Practical coding examples implementing concepts from the lecture. - Code examples - Implementation details - Practical applications - Coding exercises

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NOTATION CONVENTIONS¶

Category	Notation
SCALARS	\(a, b, c, \alpha, \beta, \gamma\)
VECTORS	\(\mathbf{x}, \mathbf{y}, \mathbf{z}\)
MATRICES	\(\mathbf{X}, \mathbf{Y}, \mathbf{Z}\)
SETS	\(A, B, C\)
NUMBER SYSTEMS	\(\mathbb{R}, \mathbb{C}, \mathbb{Z}, \mathbb{N}, \mathbb{R}^n\)
PROBABILITY	\(p(\cdot), P[\cdot]\)

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