18.6 Superposition of Solutions for HLSP

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: N/A

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Superposition of Solutions for HLSP

Suppose $A \in R^{m \times n}$ is "given." Suppose $A$ has a total of $0 \leq d \leq n$ nonpivot columns. Let ${\vec{z}}_{1}, {\vec{z}}_{2}, \dots, {\vec{z}}_{d}$ be linearly independent solutions to the homogenous linear equation $A \cdot \vec{x} = \vec{0}$ . In other words, let

A \cdot {\vec{z}}_{i} = \vec{0}

for each $i \in {1, 2, \dots, d}$ . Then, any 6.1 Linear Combination of Vectors

\vec{z} = c_{1} {\vec{z}}_{1} + c_{2} {\vec{z}}_{2} + \dots + c_{d} {\vec{z}}_{d}

is also a solution to the HLSP.