The above figure summarizes some of the interactions between the four fundamental matrix subspaces for a real matrix including whether the spaces in question are subspaces of
or ,
which subspaces are orthogonal to one another,
and how the matrix maps various vectors relative to the subspace in which lies.

In the event that , all four of the fundamental matrix subspaces are lines
in .
In this case, one can write for some vectors , whereby the directions of
the four lines correspond to , , , and . An elementary fact from linear
algebra is that these directions are also represented by the eigenvectors
of
and
(Strang 2008); this is one of the reasons why the four fundamental subspaces of
are often associated with the eigenvalues and the
singular value decompositions of
and
in many presentations of the fundamental theorem of linear algebra (Strang 2012).