Before we can effectively design algorithms to solve linear systems, we must understand the fundamental geometric and mathematical properties of the matrices that represent them.
In this section, we will explore:
Solvability: When does a system have a unique solution, infinite solutions, or no solution at all?
Norms: How do we mathematically measure the “size” and stretching power of a matrix?
Conditioning: How sensitive is a matrix to tiny uncertainties or noise in our data?
Matrix types and sparsity: What special matrix structures exist and why do they matter?
Understanding these theoretical foundations will allow us to evaluate the stability and efficiency of the numerical methods we develop in the following sections.