Editing Cauchy–Binet formula (section)

== Geometric interpretations ==

If ''A'' is a real ''m''&times;''n'' matrix, then det(''A''&nbsp;''A''<sup>T</sup>) is equal to the square of the ''m''-dimensional volume of the [[Parallelepiped#Parallelotope|parallelotope]] spanned in '''R'''<sup>''n''</sup> by the ''m'' rows of ''A''. Binet's formula states that this is equal to the sum of the squares of the volumes that arise if the parallelepiped is orthogonally projected onto the ''m''-dimensional coordinate planes (of which there are <math>\tbinom nm</math>).

In the case ''m''&nbsp;=&nbsp;1 the parallelotope is reduced to a single vector and its volume is its length. The above statement then states that the square of the length of a vector is the sum of the squares of its coordinates; this is indeed the case by [[Euclidean distance|the definition]] of that length, which is based on the [[Pythagorean theorem]].

In [[tensor algebra]], given an [[inner product space]] <math>V</math> of dimension ''n'', the Cauchy–Binet formula defines an induced inner product on the [[Exterior algebra#Inner product|exterior algebra]] <math>\wedge^m V</math>, namely:<blockquote><math>\langle v_1\wedge\cdots \wedge v_m, w_1\wedge\cdots \wedge w_m\rangle
=\det\left( \langle v_i,w_j\rangle \right)_{i,j=1}^m .</math></blockquote>