Linear systems of equations and vector spaces

Question

I'm looking for references that explicitly (and in an accesible way: -I come from engineering-) handle the connection between solving a linear system of equations and the abstract geometry involved.

Answer 1

The relation between rank and nullity is $\;\dim\ker A+\dim\operatorname{Im}A=\dim \mathbf R^{62}$, the hypotheses imply $\dim\ker A=11$.

As for the result with Maple, you should check your data: if $\dim \ker A=11$, the solutions should depend on $11$ parameters, not $10$.