Section 1: Core Linear System Theory
Overview
Focus on fundamental algebraic structures and solution spaces.
Topics Covered
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 4: Orthogonality
- Orthogonality of the Four Subspaces
- Projections
- Least squares approximations
- Orthonormal bases and Gram-Schmidt
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