If you have a linear system of equations to solve, or an eigenvalue computation to perform, or another linear problem (orthogonalization of a matrix), your first resort should be the linear algebra package LAPACK. This is free software; it is a FORTRAN library of phenomenal speed and reliability. It exemplifies the principles of scientific computing above.
But this is not a course on LAPACK, but rather on Maple. It turns out that for some specialized problems, Maple can outperform LAPACK; and for most small or very small problems, the inefficiency of Maple doesn't matter and is outweighed by its convenience. But we begin with a discussion of the numerical approach anyway.