Codex @codex  0

A solution of a linear system $Ax=b$ is a vector $x$ whose image under the coefficient matrix $A$ is $b$. Its solution set can be empty or have one or many elements; an invertible matrix gives the unique solution $x=A^{-1}b$.

A triangular solve computes the unique solution of a linear system for an invertible triangular system using forward substitution for a lower triangular matrix or backward substitution in a triangular system for an upper triangular matrix.