18.3 Homogenous Linear System

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 17.1 The General Linear-Systems Problem

Properties: 18.4 Solution to HSLP using RREF
Sufficiencies: N/A
Questions: N/A

Homogenous Linear System

If $A \in R^{m \times n}$ , then the corresponding homogenous linear-systems problem is to find all unknown vectors $\vec{x} \in R^{n}$ such that

A \cdot \vec{x} = \vec{0} .

Remark. The homogenous linear-systems problem is a special case of the 17.1 The General Linear-Systems Problem where the $m \times 1$ vector $\vec{b}$ on the right-hand side is zero.

! Note that homogenous linear-systems problems have at least one solution.

For $\vec{z} = \vec{0} \in R^{n}$ , we see that

\begin{aligned} A \cdot \vec{z} & = [\begin{array}{c} A (:, 1) & A (:, 2) & \dots & A (:, n) \end{array}] \cdot [\begin{array}{c} 0 \\ 0 \\ ⋮ \\ 0 \end{array}] \\ = \sum_{k = 1}^{n} 0 \cdot A (:, k) = \vec{0} \in R^{m} \end{aligned}

To find any nonzero solution to a HLSP, we look for linearly dependent columns of the matrix $A$ . These columns are more easily identified by transforming matrix $A$ into RREF.
$⧫$