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).
