SOLUTION SETS for LINEAR SYSTEMS

Homogeneous Systems

• homogeneous system $A{\bf x}={\bf0}$
• solution types trivial ${\bf x}= {\bf0}$, otherwise nontrivial
• $A{\bf x}={\bf0}$ has a nontrivial solution iff there is at least one free variable
• solution is in parametric form when it is written as a linear combination of vectors

Nonhomogeneous Systems

• nonhomogeneous system $A{\bf x}={\bf b}\neq {\bf0}$
• complete solution in terms of particular and homogeneous solutions:
  if $A{\bf p}= {\bf b}$ and $A{\bf v}= {\bf0}$ then ${\bf x}= {\bf p}+ {\bf v}$ is also a solution.
• geometry: if the general solution is ${\bf x}= {\bf p}+ t{\bf v}$, this is the line through ${\bf p}$ parallel to ${\bf v}$.

Writing a Solution Set

1.

Row reduce the augmented matrix

2.

Express the basic variables in terms of free variables

3.

Write ${\bf x}$ in terms of constants and free variables

4.

Write ${\bf x}$ as a linear combination of vectors (parametric form)

LINEAR INDEPENDENCE

Independence

• The set $\{ {\bf v}_1, {\bf v}_2, \dots, {\bf v}_k \}$ is linearly independent if
  
  $$x_1{\bf v}_1 + x_2{\bf v}_2 + \cdots + x_k{\bf v}_k = {\bf0}$$
  
  
  has only the trivial solution.
• The set $\{ {\bf v}_1, {\bf v}_2, \dots, {\bf v}_k \}$ is linearly dependent if there exist $c_1, c_2, \ldots, c_k$, not all zero, such that
  
  $$c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_k{\bf v}_k = {\bf0}.$$
  
  

Matrix Column Independence

The columns of A are linearly independent iff $A{\bf x}={\bf0}$ has only the trivial solution.

Characterization

$\{ {\bf v}_1, {\bf v}_2, \dots, {\bf v}_k \}$ are linearly dependent if some ${\bf v}_i$ is a linear combination of the other vectors.

Simple Checks for Dependence

• $\{ {\bf v}_1, {\bf v}_2, \dots, {\bf v}_k \} \in R^n$ are linearly dependent if k > n.
• $S=\{ {\bf v}_1, {\bf v}_2, \dots, {\bf v}_k \}$ is linearly dependent if ${\bf0}\in S$.