Solving Linear Systems with No Solution or Infinitely Many Solutions
Introduction
Addressing how to solve linear systems with no solutions or infinitely many solutions.
Previous examples had unique solutions.
Solving a Linear System with No Solutions
Example Approach:
Start by throwing the system into an augmented matrix.
Perform row operations to reach row echelon form.
Zero out entries below leading entries using operations like row 2 minus double row 1, etc.
Key Signs of No Solution:
If row reduction results in a row where all coefficients are zero but the right-hand side is non-zero (e.g., 0x + 0y + 0z = -42/5), the system has no solution.
This inconsistency (0 = non-zero) indicates no solution.
Solving a Linear System with Infinitely Many Solutions
Challenges:
Not only solving but describing the solutions since they are infinite.
Example Approach:
Use reduced row echelon form to read off solutions.
Focus on leading entries and corresponding columns (variables).
Solve for leading variables in terms of free variables.
Terms and Definitions
Leading Variables: Variables in columns with leading entries.
Free Variables: Variables in columns without leading entries.
Free variables are unrestricted, leading to infinite solutions.
Describing Solutions
Example:
Given an augmented matrix already in reduced row echelon form, rearrange to express leading variables in terms of free variables.
Example: v = 3 + 2w - 2z, x = 7 + 3z, y = 4 - z.
All solutions can be described in terms of free variables w and z.
Every solution vector can be expressed with free variables deciding values.
Example with Calculation to Find Reduced Row Echelon Form
Process:
Convert system to augmented matrix and perform row operations to reduce.
Once in reduced row echelon form, identify and rearrange equations:
Leading variables (e.g., w, y) expressed in terms of free variables (e.g., x, z).
Describe the solution vector in terms of free variables.
Dimensionality of Solution Set
Number of free variables corresponds to solution set dimensionality.
Example: Two free variables imply a 2-dimensional solution set, forming a plane in 4-dimensional space.
Conclusion
Understanding how to solve and describe solutions for systems with no solutions or infinitely many solutions.
Completion of week 5, full understanding of solving linear systems.