Linear Algebra — Basic

1. Chapter 1 Fundamentals of Vector Spaces

   Axiomatic definition, beyond $\mathbb{R}^n$ — polynomials and functions as vectors

2. Chapter 2 Linear Independence and Basis

   Linear combinations, linear independence, basis, dimension

3. Chapter 3 Introduction to Determinants: Cramer's Rule

   Deriving the determinant from systems of equations — a historical approach

4. Chapter 4 Determinants: Cofactor Expansion

   Cofactors, minors, Laplace expansion, inverse matrices

5. Chapter 5 Visual Understanding of Determinants

   Shear transformations, parallelograms, changes in area and volume

6. Chapter 6 Eigenvalues and Eigenvectors

   Definition, geometric meaning, characteristic polynomial, eigenspaces

7. Chapter 7 Properties and Applications of Eigenvalues

   Trace and determinant, triangular matrices, linear independence, applications

8. Chapter 8 Structure of Solutions of Linear Systems

   Homogeneous and non-homogeneous systems, solution space structure theorem, rank-nullity theorem

9. Chapter 9 The Identity Matrix

   Definition and basic properties, eigenvalues, the identity element of matrix multiplication

10. Chapter 10 The Kronecker Delta

    Definition of δ_ij, Einstein notation, foundations of tensor notation

11. Chapter 11 Matrix Rank

    Definition of rank, pivot-based computation via row reduction, rank-nullity theorem, full-rank equivalences. Appendix: Strang-style proof of row rank = column rank

12. Chapter 12 Permutation Matrices

    Definition, orthogonality, determinant, relation to LU decomposition, symmetric group, applications

Related Topics (Basic Special Matrices & Concepts)

• Adjoint Matrix

  Cofactor matrix adj(A), A·adj(A) = det(A)·I, inverse A^{-1} = adj(A)/det(A), distinction from Hermitian adjoint

• Change of Basis

  Change-of-basis matrix, transformation of coordinate vectors, similarity transformation A' = P^{-1}AP, relation to diagonalization

• Cross Product

  Vector product in R³. Determinant form, normal vector, parallelogram area, triple product, generalization to exterior algebra

• Diagonal Matrix

  Ease of operations on diag(d_1,...,d_n) (multiplication, inverse, powers), relation to diagonalization A=PDP^{-1}

• Direct Sum

  Internal direct sum V=W₁⊕W₂ with W₁∩W₂={0}, external direct sum, relation to projections, applications to Jordan decomposition

• Gram-Schmidt Orthonormalization

  Classical and modified GS algorithms, R³ examples, relation to QR decomposition, numerical stability

• Orthogonal Matrix

  Definition Q^TQ=I, properties, relation to rotation matrices, Gram-Schmidt orthogonalization, applications

• Orthonormal Basis

  ⟨e_i, e_j⟩ = δ_{ij}, Fourier coefficients for coordinates, Parseval's identity, relation to QR decomposition

• Rank-Nullity Theorem

  Proof of dim V = rank T + nullity T, row rank = column rank, surjectivity/injectivity conditions for linear maps

• Skew-Symmetric Matrix

  A^T=-A, zero diagonal, purely imaginary eigenvalues, relation to cross product [a]×

• Symmetric Matrix

  A^T=A, spectral theorem (real symmetric matrices are diagonalizable by an orthogonal matrix), real eigenvalues, relation to quadratic forms

• Triangular Matrix

  Upper/lower triangular definitions, determinant = product of diagonal entries, eigenvalues = diagonal entries, relation to LU decomposition, forward/back substitution

• Unitary Matrix

  Complex analog of orthogonal matrices U*U=I, properties, eigenvalues with absolute value 1, applications to quantum computing

• Vandermonde Matrix

  Product formula for the determinant, polynomial interpolation, applications to Reed-Solomon codes