Linear Algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces, linear transformations, and systems of linear equations. It provides a framework for modeling and manipulating relationships that are "linear," meaning they can be represented geometrically as lines, planes, or their higher-dimensional analogs called hyperplanes. The primary tools of linear algebra are matrices and vectors, which offer a powerful way to represent data, solve complex systems of equations, and describe transformations like rotations, scaling, and shears in space, making it a foundational tool in fields ranging from computer graphics and data science to physics and engineering.

1. Foundations of Linear Systems
   1. Introduction to Linear Equations
      1. Definition of a linear equation
      2. Standard form of a linear equation
      3. Variables and coefficients
      4. Solutions and solution sets
      5. Geometric interpretation in two dimensions
      6. Geometric interpretation in three dimensions
   2. Systems of Linear Equations
      1. Definition of a linear system
      2. Representation as a set of equations
      3. Systems of two linear equations in two variables
      4. Systems of three linear equations in three variables
      5. Consistent and inconsistent systems
      6. Homogeneous and nonhomogeneous systems
      7. Types of solution sets
         1. Unique solutions
         2. Infinitely many solutions
         3. No solution
      8. Geometric interpretation of solution sets
   3. Methods of Solving Linear Systems
      1. Substitution method
      2. Elimination method
      3. Gaussian elimination
         1. Elementary row operations
         2. Row echelon form
         3. Back-substitution process
      4. Gauss-Jordan elimination
         1. Reduced row echelon form
         2. Interpreting solutions from RREF
      5. Matrix representation of systems
         1. Coefficient matrix
         2. Augmented matrix
         3. Matrix form of linear systems
      6. Solution sets in parametric vector form