Notes from Lecture 2 - Linear Algebra

Jul 30, 2024

Lecture 2: Linear Algebra

Main Topics

  • Example system of equations
  • Method of solution: Elimination
  • Discussion of matrix concepts

System of Equations

  • Goal: Solve the system using elimination, not determinants (determinants will be covered later).
  • Matrix format: Ax = b where A is a matrix, x is a vector of unknowns, and b is a result vector.

Elimination Method

  • Forward Elimination: Steps to eliminate variables.
  • Back Substitution: Once in triangular form, solve for unknowns from the last back up to the first.

Steps of Forward Elimination

  1. Identify Pivot - Start with the first equation (First pivot).
  2. Perform Multiplications - Use multipliers to eliminate variables in subsequent equations.
    • Example: For row 2, multiply row 1 by a suitable factor to eliminate x terms from row 2.
  3. Update Matrix - Keep the results in a new augmented matrix format so right-hand side values are updated.

Matrix Representation of Elimination

  • Express elimination as matrix operations:
    • Keep track of transformations applied to the matrix A.
    • Define elementary matrices for each operation.
    • The objective: Obtain upper triangular matrix U from A using elimination matrices (E).

Elementary/Elimination Matrices

  • For specific elimination steps;
    • E21 for eliminating terms from row 2 based on row 1.
    • E32 for eliminating terms from row 3 based on row 2.
  • Example structure of elementary matrix:
    • E21 =
      [ 1 0 0 ]
      [ -3 1 0 ]
      [ 0 0 1 ]
    • Have 1 for unchanged rows and respective coefficient for the operation.

Detecting Failure in Elimination

  • Possible points of failure during elimination:
    • Zero in pivot position could indicate trouble (e.g., rows may need to be exchanged).
    • If cannot find a non-zero entry below the problematic pivot after an exchange, it indicates failure to complete elimination.
  • Strategies for handling zeros:
    • Row exchange to find a proper pivot above.

Back Substitution

  • Once in an upper triangular form, solve for variables in reverse order.
  • Can solve for z first, then substitute back to solve for y, and subsequently solve for x.

Matrix Multiplication and Transformations

  • Important properties of matrix operations:
    • Order of matrices matters (non-commutative property).
    • Ability to change parentheses due to associative property.
  • Matrices allow operations on rows or columns depending on the side of multiplication:
    • Left multiplication affects rows (elimination)
    • Right multiplication affects columns (not covered in detail yet).

Overview of Inverse Operations

  • Inverses of elementary matrices allow reverting back to the original matrix; the process of undoing elimination steps.
    • For example: If elimination subtracts, the inverse would add back the respective values.

Summary

  • Today’s focus on method of elimination for solving systems represented by matrices.
  • Integral to understanding operations leading towards upper triangular form (U).
  • Basis for exploring concepts of inverse matrices next session.

Final Thoughts

  • Elaborated on error handling during elimination.
  • Ready to dive deeper into inverse matrices next lecture.