🔹 1/2 Page (Concept Capsule)

Use for: Definitions, remarks, properties, short results

Recommended layout

  • 1–2 sentence contextual opening
  • Precise definition or statement
  • 1 short example (preferably physical)
  • Key formula/result (boxed or highlighted)

Typical content

  • No proofs
  • No derivations
  • At most 1 equation

🔹 1 Page (Core Concept Unit)

Use for: Fundamental ideas, basic tools

Recommended layout

  • Short intro paragraph (why it matters in physics)
  • Formal definition(s)
  • Mathematical formulation
  • 1 worked example
  • 1–2 important remarks/properties

Typical content

  • 2–4 equations
  • Minimal derivation
  • One simple diagram if needed

🔹 2 Pages (Concept + Method)

Use for: Standard techniques, theorems without full rigor

Recommended layout

  • Motivation from physics
  • Definitions and mathematical setup
  • Step-by-step method or theorem
  • 1 detailed worked example
  • Physical interpretation
  • Common mistakes / notes

Typical content

  • 5–8 equations
  • Partial derivation
  • Clear algorithmic steps

🔹 3 Pages (Core Theoretical Block)

Use for: High-weight syllabus topics

Recommended layout

  • Physical motivation and context
  • Formal theory and definitions
  • Key theorem(s)
  • Proof outline (not fully abstract)
  • 2 worked examples (math + physics)
  • Interpretation and summary box

Typical content

  • 8–12 equations
  • One diagram or schematic
  • Short proof or logical derivation

🔹 4 Pages (Major Exam Topic)

Use for: Eigenvalues, Hermitian operators, vector spaces

Recommended layout

  • Real physical problem as introduction
  • Mathematical framework
  • One major theorem with derivation
  • Supporting lemmas/results
  • 2–3 worked examples
  • Applications to physics
  • Quick revision points

Typical content

  • 12–18 equations
  • One full derivation
  • Exam-oriented remarks

🔹 5 Pages (Core + Applications)

Use for: Central chapters bridging math and physics

Recommended layout

  • Physical motivation (problem-based)
  • Definitions and theory
  • Complete derivation or theorem
  • 3 worked examples
  • Application to at least two physics areas
  • Summary and formula sheet

Typical content

  • 18–25 equations
  • Diagrams/graphs where applicable
  • Concept–exam balance

🔹 6 Pages (Foundational / Heavy-Weight Topics)

Use for: Eigenvalue problems, inner product spaces, operators

Recommended layout

  • Physical background and motivation
  • Mathematical development (stepwise)
  • One major theorem with proof
  • Supporting results
  • 3–4 worked examples
  • Physical interpretation (QM/CM)
  • Common pitfalls
  • Chapter mini-summary

Typical content

  • 25–35 equations
  • Multiple derivations
  • High UGC–NET/JAM relevance

🧠 Summary Table

Page Length Purpose Depth
1/2 page Concept / definition Very light
1 page Basic concept Light
2 pages Method / tool Moderate
3 pages Core theory Strong
4 pages Major topic Deep
5 pages Core + applications Very deep
6 pages Foundational block Maximum

Linear Algebra for Physicists

Chapter 1. Introduction to Linear Algebra in Physics (≈ 10 pages)

  • Role of linear algebra in physical theories (1/2 page)

  • Scalars and vectors as physical quantities (1 page)

  • Real and complex number systems (2 pages)
    • Algebra of real numbers (1 page)
    • Complex numbers, conjugation, modulus (1 page)
  • Summation notation and index representation (2 pages)
    • Einstein summation convention (1 page)
    • Kronecker delta and Levi-Civita symbol (1 page)
  • Physical examples motivating vector spaces (4.5 pages)
    • Displacement, velocity, force vectors (1.5 pages)
    • State vectors in quantum mechanics (1.5 pages)
    • Function spaces in physics (1.5 pages)

Chapter 2. Matrices and Determinants (≈ 30 pages)

  • Matrices and their physical interpretation (2 pages)
    • Matrix representation of physical systems (1 page)
    • Coordinate transformations (1 page)
  • Types of matrices (6 pages)
    • Diagonal and triangular matrices (1.5 pages)
    • Symmetric and antisymmetric matrices (1.5 pages)
    • Hermitian matrices (1.5 pages)
    • Unitary matrices (1.5 pages)
  • Matrix operations (5 pages)
    • Addition and scalar multiplication (1 page)
    • Matrix multiplication (2 pages)
    • Transpose and conjugate transpose (2 pages)
  • Determinants: definition and properties (6 pages)
    • Definition and expansion (2 pages)
    • Properties of determinants (2 pages)
    • Determinants of special matrices (2 pages)
  • Determinants and invertibility (3 pages)
    • Singular and non-singular matrices (1.5 pages)
    • Physical interpretation (1.5 pages)
  • Jacobians and coordinate transformations (4 pages)
    • Jacobians in variable transformation (2 pages)
    • Applications in mechanics and EM (2 pages)
  • Rank of a matrix and its significance (4 pages)
    • Rank and minors (2 pages)
    • Rank and physical constraints (2 pages)

Chapter 3. Systems of Linear Equations (≈ 18 pages)

  • Linear systems arising in physics (3 pages)
    • Circuit equations (1.5 pages)
    • Coupled mechanical systems (1.5 pages)
  • Homogeneous and non-homogeneous systems (3 pages)
    • Physical interpretation (1.5 pages)
    • Existence of solutions (1.5 pages)
  • Matrix representation of linear systems (2 pages)
    • Augmented matrices (1 page)
    • System equivalence (1 page)
  • Elementary row operations (4 pages)
    • Row transformations (2 pages)
    • Preservation of solutions (2 pages)
  • Row-reduced echelon form (3 pages)
    • Algorithm (1.5 pages)
    • Interpretation of solutions (1.5 pages)
  • Consistency and solution criteria (3 pages)
    • Rank conditions (1.5 pages)
    • Physical meaning (1.5 pages)

Chapter 4. Vector Spaces (≈ 20 pages)

  • Definition of vector spaces (2 pages)
    • Axioms (1 page)
    • Examples from physics (1 page)
  • Examples from classical and quantum physics (4 pages)
    • Geometrical vectors (2 pages)
    • Quantum state spaces (2 pages)
  • Subspaces and their properties (4 pages)
    • Criteria for subspaces (2 pages)
    • Physical constraints (2 pages)
  • Linear dependence and independence (4 pages)
    • Definitions and tests (2 pages)
    • Physical interpretation (2 pages)
  • Basis and dimension (4 pages)
    • Choice of basis (2 pages)
    • Degrees of freedom (2 pages)
  • Coordinate representation of vectors (2 pages)
    • Component form (1 page)
    • Change of coordinates (1 page)

Chapter 5. Linear Transformations and Operators (≈ 20 pages)

  • Linear transformations in physical problems (3 pages)
    • Rotations and reflections (1.5 pages)
    • Operators in mechanics (1.5 pages)
  • Kernel and range (4 pages)
    • Null space (2 pages)
    • Image and range (2 pages)
  • Rank–nullity theorem (2 pages)
    • Statement and proof (1 page)
    • Physical interpretation (1 page)
  • Matrix representation of linear transformations (5 pages)
    • Transformation matrices (2 pages)
    • Operator representation in different bases (3 pages)
  • Change of basis (3 pages)
    • Similarity transformations (1.5 pages)
    • Physical invariance (1.5 pages)
  • Composition of linear operators (3 pages)
    • Operator algebra (1.5 pages)
    • Physical examples (1.5 pages)

Chapter 6. Eigenvalues and Eigenvectors (≈ 26 pages)

  • Eigenvalue problems in physics (4 pages)
    • Definition and examples (2 pages)
    • Physical significance (2 pages)
  • Characteristic equation (4 pages)
    • Characteristic polynomial (2 pages)
    • Roots and multiplicity (2 pages)
  • Cayley–Hamilton theorem (4 pages)
    • Proof (2 pages)
    • Applications (2 pages)
  • Diagonalization of matrices (6 pages)
    • Conditions for diagonalization (3 pages)
    • Physical systems (3 pages)
  • Degeneracy and physical interpretation (4 pages)
    • Degenerate eigenvalues (2 pages)
    • Symmetry considerations (2 pages)
  • Applications to coupled oscillations (4 pages)
    • Normal mode analysis (2 pages)
    • Frequency spectrum (2 pages)

Chapter 7. Inner Product Spaces (≈ 18 pages)

  • Inner products (3 pages)
    • Real vector spaces (1.5 pages)
    • Complex vector spaces (1.5 pages)
  • Norms and orthogonality (3 pages)
    • Norm and distance (1.5 pages)
    • Orthogonality conditions (1.5 pages)
  • Orthonormal bases (3 pages)
    • Construction (1.5 pages)
    • Physical meaning (1.5 pages)
  • Gram–Schmidt orthogonalization (4 pages)
    • Algorithm (2 pages)
    • Applications (2 pages)
  • Projection operators (3 pages)
    • Definition (1.5 pages)
    • Physical applications (1.5 pages)
  • Completeness relations (2 pages)
    • Mathematical form (1 page)
    • Use in quantum mechanics (1 page)

Chapter 8. Hermitian, Unitary, and Normal Operators (≈ 20 pages)

  • Linear operators in quantum mechanics (3 pages)
    • Operators and observables (1.5 pages)
    • Expectation values (1.5 pages)
  • Hermitian operators (6 pages)
    • Properties (2 pages)
    • Eigenvalues and eigenvectors (2 pages)
    • Physical observables (2 pages)
  • Unitary operators (5 pages)
    • Definition and properties (2.5 pages)
    • Symmetry and time evolution (2.5 pages)
  • Normal operators (3 pages)
    • Definition (1.5 pages)
    • Relation to Hermitian and unitary (1.5 pages)
  • Commutators and simultaneous diagonalization (3 pages)
    • Commutation relations (1.5 pages)
    • Physical implications (1.5 pages)

Chapter 9. Canonical Forms of Matrices (≈ 12 pages)

  • Similarity transformations (3 pages)
    • Definition (1.5 pages)
    • Physical invariance (1.5 pages)
  • Diagonal and triangular forms (3 pages)
    • Reduction techniques (1.5 pages)
    • Examples (1.5 pages)
  • Jordan canonical form (6 pages)
    • Jordan blocks (2 pages)
    • Construction procedure (2 pages)
    • Physical relevance (2 pages)

Chapter 10. Quadratic and Bilinear Forms (≈ 10 pages)

  • Bilinear forms (2 pages)
    • Definition and examples (1 page)
    • Physical interpretation (1 page)
  • Quadratic forms (3 pages)
    • Matrix representation (1.5 pages)
    • Examples in mechanics (1.5 pages)
  • Symmetric and skew-symmetric forms (3 pages)
    • Properties (1.5 pages)
    • Applications (1.5 pages)
  • Reduction to canonical form (2 pages)
    • Diagonalization (1 page)
    • Stability analysis (1 page)

Chapter 11. Applications in Physics (≈ 16 pages)

  • Normal modes of vibrations (4 pages)
    • Two-body systems (2 pages)
    • Multi-degree systems (2 pages)
  • Small oscillations and coupled systems (3 pages)
    • Linearization (1.5 pages)
    • Physical interpretation (1.5 pages)
  • Rotations and angular momentum matrices (3 pages)
    • Rotation matrices (1.5 pages)
    • Angular momentum operators (1.5 pages)
  • Matrix formulation of Schrödinger equation (4 pages)
    • Operator form (2 pages)
    • Eigenvalue interpretation (2 pages)
  • Operator methods in quantum mechanics (2 pages)
    • Expectation values (1 page)
    • Measurement theory (1 page)