Ch 1.5 - Solution Sets of Linear Systems

Flashcards covering key vocabulary terms and definitions from the lecture notes on Solution Sets of Linear Systems.

Homogeneous Linear System

A system of linear equations that can be written in the form Ax = 0, where A is an m x n matrix and 0 is the zero vector in R_m.

Trivial Solution

The zero vector x = 0, which is always a solution to a homogeneous linear system Ax = 0.

Nontrivial Solution

A nonzero vector x that satisfies the homogeneous linear system Ax = 0.

Condition for Nontrivial Solutions (Ax=0)

The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.

Solution Set of Ax=0

Can always be expressed explicitly as the Span of a set of vectors {v1, …, vp}. If the only solution is the zero vector, the solution set is Span {0}.

Parametric Vector Form

An explicit description of a solution set using vectors, often written as x = su + tv or x = tv, where parameters vary over all real numbers.

Solution of a Nonhomogeneous System (Ax=b)

If consistent, the general solution is written in parametric vector form as a particular solution vector (p) plus an arbitrary linear combination of vectors that satisfy the corresponding homogeneous system (Ax = 0).

Theorem 6 (Solution Set Relationship)

If Ax=b is consistent for some given b, and p is a solution, then the solution set of Ax=b is the set of all vectors of the form w = p + vh, where vh is any solution of the homogeneous equation Ax = 0.

Steps to Write Solution Set in Parametric Vector Form

1. Row reduce augmented matrix to reduced echelon form. 2. Express basic variables in terms of free variables. 3. Write typical solution x as a vector dependent on free variables. 4. Decompose x into a linear combination of vectors using free variables as parameters.

Solution Sets of Linear Systems

Important objects of study in linear algebra that can be described explicitly and geometrically using vector notation.