Section 1.5: Solution Sets Of Linear Systems

Homogeneous Linear Systems

A system of linear equations is defined as homogeneous if it can be written in the form:

$Ax = 0$

Where:

• $A$ is an $m \times n$ matrix.

• $0$ is the zero vector in $\mathbb{R}^m$.

Solutions of Homogeneous Systems:

• The Trivial Solution: A homogeneous system $Ax = 0$ always has at least one solution, specifically $x = 0$ (the zero vector in $\mathbb{R}^n$).

• Nontrivial Solutions: The homogeneous equation $Ax = 0$ has a nontrivial solution (a non-zero solution) if and only if the system has at least one free variable.

Example 1: Describing a Homogeneous Solution Set

• Objective: Determine if a system has a nontrivial solution and describe the solution set.

• Procedure:

  • Let $A$ be the matrix of coefficients.

  • Row reduce the augmented matrix $[A \mid 0]$ to echelon form.

  • In this example, $x_3$ is identified as a free variable, which implies the existence of nontrivial solutions (one for each choice of $x_3$).

  • Continue the row reduction to reduced echelon form.

  • Solve for the basic variables $x_1$ and $x_2$ in terms of the free variable $x_2$.

• Vector Representation:

  • The general solution can be written as a vector: $x = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$.

  • This vector is factored to express it as a scalar multiple of a specific vector $v$: $x = x_3 v$.

• Geometric Context: This shows that every solution is a scalar multiple of $v$. The trivial solution occurs when $x_3 = 0$. Geometrically, the solution set is a line passing through the origin ($0$) in $\mathbb{R}^3$.

Parametric Vector Form

Definitions and representations of solution sets using vectors are as follows:

• Parametric Vector Equation of a Line: An equation in the form $x = tv$ (with $t$ in $\mathbb{R}$). In Example 1, the solution $x = x_3 v$ is a parametric vector equation of a line.

• Parametric Vector Equation of a Plane: An equation in the form $x = s u + t v$ (where $s$ and $t$ are parameters).

• Definition: Whenever a solution set is described explicitly using vectors, the solution is said to be in parametric vector form.

Solutions of Nonhomogeneous Systems

When a nonhomogeneous linear system ($Ax = b$, where $b \neq 0$) has many solutions, the general solution is expressed as a particular vector plus an arbitrary linear combination of vectors that satisfy the corresponding homogeneous system.

Example 2: Describing Solutions of $Ax = b$

• Row operations on the augmented matrix $[A \mid b]$ produce a reduced form where basic variables are expressed in terms of free variables.

• For this example, $x_3$ is free.

• The general solution is expressed in the form:

$x = p + x_3 v$

• Here, $p$ is a specific vector representing a particular solution to $Ax = b$, and $x_3 v$ represents the general solution to the homogeneous equation $Ax = 0$.

• Writing $t$ as a general parameter, the solution set is described by $x = p + tv$.

• Verification: Correctness can be verified by checking that $A(p + tv) = b$.

Geometric Interpretation and Translation

The relationship between solutions for $Ax = b$ and $Ax = 0$ can be understood via the concept of translation in $\mathbb{R}^n$.

• Translation Logic: Given vectors $v$ and $p$, adding $p$ to $v$ translates $v$ in a direction parallel to the line through $p$ and $0$. We say $v$ is translated by $p$ to $v + p$.

• Effect on Lines: If each point on a line $L$ in $\mathbb{R}^n$ is translated by a vector $p$, the result is a line parallel to $L$.

• Application to Systems:

  • The solution set of $Ax = 0$ is a line $L$ through the origin ($0$) and vector $v$.

  • Adding the particular solution $p$ to each point on $L$ produces the translated line defined by $x = p + tv$.

  • The solution set of $Ax = b$ is thus a line through $p$ parallel to the solution set of $Ax = 0$.

Theorem 6: Structure of the General Solution

Theorem Statement: Suppose the equation $Ax = b$ is consistent for some given $b$, and let $p$ be a particular solution. Then the solution set of $Ax = b$ is the set of all vectors of the form:

$w = p + v_h$

Where $v_h$ is any solution of the corresponding homogeneous equation $Ax = 0$.

Key takeaway: If $Ax = b$ has a solution, the entire solution set is obtained by translating the solution set of the homogeneous system $Ax = 0$ using any particular solution $p$ as the translation vector. This generalizes to larger solution sets (like planes) when multiple free variables are present.

Procedure For Writing Solution Sets in Parametric Vector Form

To write the solution set of a consistent system in parametric vector form, use the following steps:

1. Row Reduction: Reduce the augmented matrix to reduced echelon form.

2. Basic Variable Expression: Express each basic variable in terms of any free variables appearing in the system's equations.

3. General Vector Formulation: Write a typical solution vector $x$ whose entries depend on the identified free variables.

4. Decomposition: Decompose the vector $x$ into a linear combination of vectors (containing numeric entries) using the free variables as parameters.