Is there a text which treats in depth linear subspaces of all dimensions in $n$-dimensional space? Best representations, formulas for intersections and rotations etc.
EDIT: What I am interested in most is the best representation of the subspaces. They are often represented with arbitrary bases. I need some "universal" representation (basis-free?), when we can in one moment say that two subspaces is the same space, without calculations.
EDIT 2: I have one idea about natural representation of linear subspaces without basis. Example in 3 dimensions: sphere ($x^2 + y^2 + z^2 = r^2$) we can squeeze to point, cylinder ($x^2 + y^2 = r^2$) - to line, two planes($x^2 = r^2$) - to plane. So line with vector $(1, 1, 1)$ can be represented with formula $(x-y)^2 + (y-z)^2 + (z-x)^2$. This can be generalized to higher dimensions. Is this idea occured somewhere already?