The fundamental matrix subspaces of a real 
matrix 
are four associated vector subspaces,
namely the column spaces and the null
spaces of the matrices 
and its transpose 
. These four subspaces
are important for a number of reasons, one of which is the crucial role they play
in the so-called fundamental theorem
of linear algebra.
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 pairs of subspaces are orthogonal
subspaces, and how the matrix 
maps various vectors 
relative to the subspace in which
lies.
When 
and 
has matrix rank 1, 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).
