2B The General Linear-Systems Problem

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: What do quadratic equations teach us about general linear systems?
Constructions: N/A
Generalizations: 2A The Nonsingular Linear-Systems Problem

Properties: The Three Types of Solutions to General Linear Systems
Sufficiencies: N/A
Questions: N/A

Problem 2B: The General Linear-Systems Problem

Let $m$ , $n \in N$ . Let $A \in R^{m \times n}$ be a given rectangular matrix and $\vec{b} \in R^{m}$ be a "given" vector. Then, the general linear systems problem is to find an unknown vector $\vec{x} \in R^{n}$ such that:

A \cdot \vec{x} = \vec{b}

The reason we break problem 2 into this and 2A The Nonsingular Linear-Systems Problem is to specify which model to use. Most of the technique used to solve general linear-systems problems are built on intuition from:

We will primarily focus on 2A The Nonsingular Linear-Systems Problem because Problem 2B can be transformed into 2A The Nonsingular Linear-Systems Problem combined with 1 The Matrix-Vector Multiplication Problem. Through this, we are transforming a difficult problem into a nonsingular-linear systems problem.