Linear Systems: Solutions, Homogeneous and Nonhomogeneous

Objectives

• Understanding the properties of linear systems
  • Homogeneous systems
  • Nonhomogeneous systems
• Steps to write solutions in parametric form

Solution Set of Linear Systems

• A solution of a linear system is a list of values (S₁, S₂,…, Sₖ) that satisfy each equation when substituted for the variables X.
• The complete collection of all possible solutions is referred to as the solution set.
• A linear system is consistent if:
  • There is a unique solution (no free variables).
  • There are infinitely many solutions (at least one free variable).
• When there is a unique solution, the solution set can be described as a single vector. When infinitely many solutions exist, this is referred to as the span of vectors.
• A system is guaranteed to be consistent if the echelon form of its coefficient matrix has a pivot in every row.
• Physical interpretation: The solutions to a linear system represent points in a physical space.

Homogeneous Systems

• A homogeneous system can be expressed as $Ax = 0$.
• Examples:
  • $-7x<em>1 + x</em>2 - 3x_3 = 0$
  • Solution: x = Spanegin{bmatrix} 0 \ 0 \ 0 \ ext{(origin)} \ ext{Trivial solution: }Span {0}
• To have a nontrivial solution, there must be at least one free variable.
• The general solution can include expressions where free variables offer a span in their vector form.

Nonhomogeneous Systems

• A nonhomogeneous system is noted as $Ax = b$, where $b <br /> eq 0$.
• Such systems can be either consistent or inconsistent.
  • Inconsistent: No solutions, typically denoted as ext{empty set} = ext{or} ext{ } ext{} .
  • Consistent: At least one solution exists, which can be nonzero.
• Example:
  • Given an augmented matrix leading to a solution $x = [ 2, 1, 3 ]$.
  • Another example leading to a vector form solution with parameters based on free variables.

Steps for Writing Solutions in Parametric Form

1. Determine the Reduced Row Echelon Form (RREF) of the augmented matrix.
2. Express each basic variable in terms of the corresponding free variables.
3. Write the solution in vector form.
4. Decompose the vector into a linear combination of vectors in parametric form.
5. Present your answer in parametric form,

Solutions and Their Representations

• The nature of a solution to the system depends on the number of free variables.
  • No free variables: Unique solution (point).
  • One free variable: Infinite solutions forming a line (span {v}).
  • Two or more free variables: Infinite solutions forming a plane or higher-dimensional space (span of multiple vectors).

Conclusion

• If $Ax = b$ is consistent, then the solution set can be represented:
  • By translating the solution set of $Ax = 0$.
  • By expressing solutions in the form of vectors that show how they build off of existing solutions to $Ax = 0$.
• Understanding the relationship between homogeneous and nonhomogeneous systems is key to solving linear equations systematically.