Linear algebra is one of the corner stones of modern computational mathematics. Almost all numerical schemes such as the finite element method and finite difference method are in fact techniques that transform, assemble, reduce, rearrange, and/or approximate the differential, integral, or other types of equations to systems of linear algebraic equations.
A system of linear algebraic equations can be expressed as
or
Solving a system with a coefficient matrix is equivalent to finding the intersection point(s) of all m surfaces (lines) in an n dimensional space. If all m surfaces happen to pass through a single point then the solution is unique. If the intersected part is a line or a surface, there are an infinite number of solutions, usually expressed by a particular solution added to a linear combination of typically vectors. Otherwise, the solution does not exist.
The core of solving a system of linear algebraic equations is decomposing the coefficient matrix. Through the decomposition process, the coupled equations are decoupled and the solution can be obtained with much less effort. A better decomposition method will perform faster and introduce less errors. Common numerical methods used to solve linear algebraic equations are briefly discussed in this section:
• Gaussian Elimination
• LU Decomposition
• SV Decomposition
• QR Decomposition
