In Exercises 5 and 6, follow the method of Examples 1 and 2...

In Exercises 5 and $6,$ follow the method of Examples 1 and 2 to write the solution set of the given homogeneous system in parametric vector form.
$$
\begin{aligned} x_{1}+3 x_{2}-5 x_{3} &=0 \\ x_{1}+4 x_{2}-8 x_{3} &=0 \\-3 x_{1}-7 x_{2}+9 x_{3} &=0 \end{aligned}
$$

Key Concepts

Homogeneous Systems

A homogeneous system of linear equations is one in which all of the constant terms are zero. Such systems always have at least the trivial solution (where all variables are zero), and if there are free variables due to dependencies among the equations, there will be infinitely many solutions that form a vector space.

Parametric Vector Form

Parametric vector form is a way of expressing the solution set of a linear system by introducing parameters for the free variables. Each solution is written as a vector that is a linear combination of constant vectors multiplied by the free parameters, making the structure of the solution space more transparent.

Free Variables

In the process of row reduction, some variables do not correspond to pivots (leading ones) in the matrix and are thus free to take any value. These free variables are assigned parameters that represent the degrees of freedom in the solution, and they play a critical role in forming the parametric vector representation.

Null Space

The null space of a matrix consists of all solutions to the homogeneous system represented by that matrix. It is a subspace of the domain and its dimension, called the nullity, indicates the number of free parameters in the general solution.

Row Reduction

Row reduction, including methods such as Gaussian elimination, simplifies the matrix associated with a system of linear equations into row echelon or reduced row echelon form. This process makes it easier to identify pivot and free variables, and it lays the groundwork for expressing the solution in parametric vector form.

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