The fundamental theorem of linear algebra for a real 
matrix 
is a collection of results relating various properties of
the four fundamental matrix subspaces
of 
.
In particular:
1. 
and 
where here, 
denotes the range or column
space of 
, 
denotes its transpose, and
denotes its null space.
2. The null space 
is orthogonal to the
row space 
.
3. There exist orthonormal bases for both the column space 
and the row space 
of 
.
4. The orthonormal bases of 
and 
can be chosen so that the restricted linear
transformation represented by 
has a diagonal matrix.
The third item on this list stems from Gram-Schmidt Orthonormalization; the fourth item stems from the singular
value decomposition of 
. Also, while different, the first item is reminiscent of the
rank-nullity theorem.
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. This diagram essentially makes visual the first
two parts of the above-stated result.
Worth noting is that this theorem is often stated differently and with varying numbers of parts. In particular, it is relatively common for versions of this theorem to include only the first two items given above, though the importance of the last two items is often cited to justify stating a four-part version like the above (Strang 1993). Some authors also include corollaries of the above statements within the statements themselves (Badger 2012).
