Given a real 
matrix 
, there are four associated vector
subspaces which are known colloquially as its fundamental 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 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).
