Types: N/A
Examples: N/A
Constructions: 2B The General Linear-Systems Problem
Generalizations: 1 The Matrix-Vector Multiplication Problem

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

Problem 2A: The Nonsingular Linear-Systems Problem

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

[A]_{n \times n} \cdot [\vec{x}]_{n \times 1} = [\vec{b}]_{n \times 1}

Claim. The NLSP is known as a "backward problem". Let $f : R^{n} \to R^{n}$ be defined as

f (\vec{x}) = A \cdot \vec{x}

The input, $\vec{x}$ is unknown and desired.
The output $\vec{b}$ is known.

Because we begin in the range and work our way towards the domain, we call this a backward problem.

Remark

We will see that nonsingular matrices are very special (perhaps we might say they are what applied linear algebraists dream about: they are very beautiful):

Rng (f) = codomain (f)

Relation with 1 The Matrix-Vector Multiplication Problem

A major similarity between matrix-vector multiplication and the nonsingular linear systems problem is that both depend on matrix-vector multiplication. The process of solving the former problem is simply to calculate a product. To solve the later problem, we “reverse engineer” our matrix-vector product in order to find input vectors that produce a given output. However, both problems involve the same underlying matrix-vector multiplication function.

f (\vec{x}) = A \cdot \vec{x}

One of the major difference between the two problems is in the assumptions of matrix $A$ . For general matrix-vector products, we only need to create a rectangular $m \times n$ matrix, where $m$ may not be equal to $n$ . However, for the NLSP, we need to create a matrix with the same number of rows as columns and this matrix must have a very special columns structure.