應用數學（一）（線性代數）

Introduction & Review Ch. 1 §1–2 Systems of linear equations, matrices, Gaussian elimination, intuition for determinants.

Week 2 – Vector Spaces Ch. 1 §3–4 Axioms of vector spaces, examples, subspaces, spanning sets.

Week 3 – Linear Independence & Bases Ch. 1 §5–6 Linear independence, basis, dimension theorem.

Week 4 – Coordinates & Change of Basis Ch. 2 §1–2 Coordinate representation, transition matrices, basis change.

Week 5 – Linear Transformations Ch. 2 §3–4 Definition, matrix representation of linear maps, kernel, range.

Week 6 – Isomorphisms & Rank–Nullity Ch. 2 §5–6 Isomorphisms, dimension arguments, Rank–Nullity Theorem.

Week 7 – Determinants Ch. 3 Cofactor expansion, properties, geometric interpretation, invertibility.

Week 8 – Midterm Exam Covers Chapters 1–3.

Week 9 – Eigenvalues & Eigenvectors Ch. 4 §1–2 Eigenvalues, eigenvectors, characteristic polynomial.

Week 10 – Diagonalization & Similarity Ch. 4 §3–4 Conditions for diagonalizability, algebraic vs. geometric multiplicity, similar matrices.

Week 11 – Complex Eigenvalues & Invariant subspace Ch. 4 §5 Complex eigenvalues of real matrices, block structure,

Week 12 – Inner Product Spaces Ch. 5 §1–2 Inner products, Cauchy–Schwarz, orthogonality, Gram–Schmidt process.

Week 13 – Orthogonal Diagonalization & Spectral Theorem Ch. 5 §3–4 Orthogonal matrices, symmetric matrices, real spectral theorem.

Jordan canonical form

Singular value decomposition

Week 16 – Final Exam Covers Chapters 4–7.