Given an matrix , the fundamental theorem of linear algebra is a collection
of results relating various properties of the four
fundamental matrix subspaces of . In particular:

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).