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Section 1: Core Linear System Theory

Overview​

Focus on fundamental algebraic structures and solution spaces.

Topics Covered​

Chapter 0: Basic Concepts​

  • Scalars and vectors
  • Inner product (dot product) and outer product
  • Vector norms (L1, L2)
  • Transpose and trace
  • Covariance and correlation matrices

Chapter 1: Introduction to Vectors​

  • Vectors and linear combinations
  • Lengths and dot products
  • Matrices

Chapter 2: Solving Linear Equations​

  • Vectors and linear equations
  • The idea of elimination
  • Elimination using matrices
  • Rules for matrix operations
  • Inverse matrices
  • Factorization A = LU
  • Transposes and permutations

Chapter 3: Vector Spaces and Subspaces​

  • Spaces of vectors
  • The nullspace of A: solving Ax = 0
  • The rank and the row reduced form
  • The complete solution to Ax = b
  • Independence, basis, and dimension
  • Dimensions of the Four Subspaces

Chapter 5: Determinants​

  • Definition and the three defining properties
  • Cofactor expansion and computation via elimination
  • Properties: det⁡(AB)=det⁡(A)det⁡(B)\det(AB) = \det(A)\det(B), det⁡(AT)=det⁡(A)\det(A^T) = \det(A)
  • Geometric interpretation: volume, orientation, Jacobian
  • Cofactors, adjugate, and Cramer's rule
  • The characteristic polynomial and connection to eigenvalues

Learning Objectives​

  • Understand vector spaces and subspaces
  • Master linear independence, basis, and dimension concepts
  • Learn the Four Fundamental Subspaces
  • Work with linear transformations
  • Perform matrix multiplication and LU decomposition
  • Compute and interpret determinants geometrically