Linear Equations Flashcards: Homogenous Systems

Study Notes

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.

Description

Dive into the concepts of solution sets for linear equations with our flashcards. This quiz covers critical terminology, including homogeneous linear systems, trivial solutions, and nontrivial solutions. Perfect for students looking to enhance their understanding of linear algebra.