Linear Equations Flashcards: Homogenous Systems

Homogeneous Linear Systems

• Defined as systems of the form Ax=0, where A is an m x n matrix and 0 is the zero vector in R^m.
• Always has at least one solution, the trivial solution x=0.

Trivial and Nontrivial Solutions

• Trivial Solution: The only solution x=0 for the homogeneous equation Ax=0.
• Nontrivial Solution: Any nonzero vector x that satisfies Ax=0, indicating the existence of other solutions.

Conditions for Nontrivial Solutions

• A homogeneous equation Ax=0 has a nontrivial solution if there is at least one free variable.

Describing a Plane

• Implicit Description: A subspace W of R^n characterized by one or more homogeneous equations. Example: 10x_1 - 3x_2 - 2x_3 = 0.
• Explicit Description: W represented parametrically as a set of all linear combinations of specified vectors, often derived from the implicit description.

Parametric Vector Equation

• Expressed as x = su + tv, where s and t are real numbers, and u and v are vectors, providing an explicit way to describe a solution set.

Translation in Vector Spaces

• In R^2, translating vector v by vector p results in v becoming v + p, effectively moving v in a direction parallel to the line through points p and origin.

Solution Set of Ax = b

• Solution set of Ax=b is a line through point p, parallel to the solution set of Ax=0, defined by the equation x = p + t*v, where t is the free variable.

Theorem 6 - Solution Sets

• If Ax=b is consistent with solution p, the complete solution set is in the form w = p + v_h, with v_h being any solution of Ax=0.
• Only applicable if Ax=b has a nonzero solution; if no solution exists, the set is empty.

Writing Solution Sets in Parametric Vector Form

• Algorithm Steps:
  • Row reduce the augmented matrix to reduced echelon form.
  • Express basic variables in terms of free variables.
  • Represent a typical solution x as a vector based on free variables.
  • Decompose x into a linear combination of numerical vectors using free variables as parameters.