Objectives: Topics
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Homogeneous systems
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Parametric vector forms of solutions to linear systems
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Section 1.5 Solution Sets of Linear Systems
Worksheet 1.5 Homogeneous Systems and Parametric Vector Forms
Objectives: Goals
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Express the solution set of a linear system in parametric vector form.
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Provide a geometric interpretation of the solution set of a linear system.
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Characterize homogeneous linear systems using free variables, span, pivots, linear combinations, and echelon forms.
Definition 1.41. Homogeneous and inhomogeneous systems.
Linear systems of the form \(A\vec{x} = \vec{0}\) are homogeneous. Linear systems of the form \(A\vec{x} = \vec{b}\) (with \(\vec{b} \neq \vec{0}\)) are inhomogeneous.
Because homogeneous systems always have the trivial solution \(\vec{x} = \vec{0}\text{,}\) the interesting question is whether they have any nontrivial solutions.
Example 1.42. Identify free variables and solution set.
Identify the free variables and the solution set of the homogeneous system:
\begin{equation*}
\begin{aligned}
x_1 + 3x_2 + x_3 \amp= 0 \\
2x_1 - x_2 - 5x_3 \amp= 0 \\
x_1 \phantom{+3x_2} - 2x_3 \amp= 0
\end{aligned}
\end{equation*}
In the previous example, we expressed the solution to the system using a vector equation. This is a parametric vector form of the solution.
In general, suppose the free variables for \(A\vec{x} = \vec{0}\) are \(x_k, \ldots, x_n\text{.}\) Then all solutions can be written in the form
\begin{equation*}
\vec{x} = x_k \vec{v}_k + x_{k+1} \vec{v}_{k+1} + \cdots + x_n \vec{v}_n,
\end{equation*}
for some vectors \(\vec{v}_k, \ldots, \vec{v}_n\text{.}\) This representation is called the parametric vector form of the solution set.
Example 1.43. Parametric vector form and geometric interpretation.
Write the parametric vector form of the solution and give a geometric interpretation.
\begin{equation*}
\begin{aligned}
x_1 + 3x_2 + x_3 \amp= 9 \\
2x_1 - x_2 - 5x_3 \amp= 11 \\
x_1 \phantom{+3x_2} - 2x_3 \amp= 6
\end{aligned}
\end{equation*}
(Note: the left-hand side is the same as in the homogeneous system example.)
