Linear Algebra Lecture 2

Main Topic: Solving Systems of Equations

Method: Elimination

• Elimination is the method of focus for solving systems of equations, rather than determinants.
• Gauss Elimination: Main technique, named after Carl Friedrich Gauss.
  • Natural idea for solving linear systems.
  • Expressed in Matrix Language for simplicity and utility in linear algebra.

Steps in Elimination

1. Initial System (Example): A matrix with three equations and three unknowns.
   • Matrix form: AX = B, where A is a coefficient matrix, X is the unknown vector, and B is the result vector.
   • Example matrix and vector:
     • A = [[1, 2, 1], [3, 8, 1], [0, 4, 1]]
     • B = [2, 12, 2]
2. Perform Elimination:
   • First Pivot: Initial focus is on top-left element.
     • Pivoting involves multiplying and subtracting to create zeros below the pivot.
     • Aim: Zero out the first column elements below the pivot.
   • Second Pivot: Next non-zero element after row operations.
     • Continue the process to zero out further elements.
   • Continue until the matrix is in Upper Triangular Form (U).
3. Failure Conditions:
   • Zero Pivot: If a pivot value is zero, the algorithm fails.
     • Row Exchange: Required to swap rows in case of a zero pivot to continue the process.
   • Non-invertible Matrix: If zero pivots cannot be avoided, the matrix is non-invertible.
4. Back Substitution:
   • Start from the last row and move upward to solve for unknowns.
   • Example calculation to solve U matrix and C vector:
     • Z = -2, Y = 1, X = 2
   • Sequence: Solve for variable in the last position first, move upwards.

Major Concepts and Tools

1. Matrix Operations: Core focus in the course.

   • Matrix Multiplication: Combining matrices in specific ways.
   • Row Operations: Operations that affect rows of the matrix.
     • Elementary Matrices (E): Simplified matrices representing single row operations.
       • Example: E21 denotes the elementary matrix for a specific row operation.
   • Permutation Matrices (P): Used to exchange rows of a matrix.
     • Generated by modifying the identity matrix to swap desired rows.

2. Associative Law in Matrix Multiplication:

   • Parentheses can be shifted (e.g., A*(B*C) equals (A*B)*C), but order of matrices must be maintained.
   • Important for efficient matrix manipulation and proofs in linear algebra.

3. Inverse Matrices:

   • Inverse Matrix (A⁻¹): Matrix that reverses effects of original matrix when multiplied together.
   • Finding Inverses: Critical for solving systems of linear equations and understanding matrix properties.
     • The inverse of elimination matrices can be computed straightforwardly by reversing operations.

Practical Tips

• Matrix Augmentation: Include the result vector as an additional column for ease of calculations.
• Matlab Approach: Perform operations on the coefficient matrix first, then apply to result vector.
• Error Handling: Be aware of how to handle zero pivots and adjust rows appropriately.
• Notational Elements: Symbols such as E21, U, C, and P represent foundational elements in solving matrix equations.

Preparation for Future Topics

• Determinants and Inverses: Will be discussed in subsequent lectures.
• Associative Properties and Higher Operations: Deeper exploration of algebraic properties of matrices.