# Grassmann Formula

I'm in my first year of Mathematics at the University. Recently, we've learnt about Grassmann Formula and when I was making a little research on the internet, I couldn't find a single reference confirming whether it is named that way because Hermann Grassmann was the one who discovered it or it is just named after him by another person. I was just very curious to know this, but I don't know where to look it up.

Does anybody know something about the explanation for the name of the formula?

For a detailed discussion of Grassmann's work see :

and see page 186 for the discussion of the formula :

After basically applying the conceptual constructions from the first section of Extension Theory in a more general way to elementary magnitudes in the second section, Graßmann introduces, after dealing with the problem of coordinates, a new form of generating products with these vector magnitudes. In quite difficult constructions, characterized by an excessive striving for extreme abstraction, he obtains a conjunctive product which is contrasted as a dual form of the exterior product and which he calls “eingewandtes Produkt” (a conjunction which later, in the 1861 revised edition of Extension Theory, is termed the “regressive product” and comprehensively developed there).

Graßmann connects his reflections to dimensional considerations on the generating system of two multivectors $A$ and $B$ entering into a multiplicative conjunction. Thus he obtains the fundamental relation:

$dim(A) + dim(B) = dim(A + B) + dim(A ∩ B)$,

that is, the sum of the dimensions of the generating systems of $A$ and $B$ is equal to the sum of the dimension of the sum-space (“umfassendes Gebiet”) and the dimension of the intersection space (“gemeinschaftliches Gebiet”) of the generating systems of $A$ and $B$ [ ref to : Linear Extension Theory. In: A new branch of mathematics. The Ausdehnungslehre of 1844 and Other Works by Hermann Graßmann. Translated by Lloyd C. Kannenberg. Chicago and La Salle: Open Court 1995, p. 202 ]. If one now equates the dimension of the initial space (“Hauptgebiet”) to $n$, then the exterior product of the multivectors $A$ and $B$ will only be unequal to zero if

$dim(A ∩ B) = 0$ and $dim(A) + dim(B) = dim(A + B) ≤ n$,

that is, when “$A$ is completely outside the domain of $B$” (hence the term “exterior product”). For the product $AB$ we can then say:

$dim(AB) = dim(A) + dim(B) = dim(A + B)$.

Of course, we have to check on the original German text : Drittes Kapitel : Das eingewandte Produkt, §125-on, to find the "best approximation" to the said formula.

See footnote page 185, with the discussion of the example of two lines (zweiter Stufe = second order):

System zweier Linien, so wird, da jene als Elementarsystem von dritter, diese von zweiter Stufe sind, das gemeinschaftliche System von $(2+2-3)$ter, d.h. von erster Stufe sein, und somit entweder durch einen Punkt [...]

Grassman identified the concept of "an algebra" in his book Die lineale Ausdehnungslehre... (The Theory of Linear Extensions). In this book he introduced the concepts used in the referenced formula - those of subspace and dimensions - as well as proving this formula.

According to this paper from the Penn State website:

He defines the notions of subspace, independence, span, dimension, join and meet of subspaces, and projections of elements onto subspaces. He is aware of the need to prove invariance of dimension under change of basis, and does so. He proves the Steinitz Exchange Theorem, named for the man who published it in 1913 (and who, incidentally, defined a linear space in terms of "units" in the same way Grassmann did). Among other such results, he shows that any finite set has an independent subset with the same span and that any independent set extends to a basis, and he proves the important identity $$\dim(U + V) = \dim(U) + \dim(V) - \dim(U \cap V).$$ He obtains the formula for change of coordinates under change of basis, defines elementary transformation of bases, and proves that every change of basis (equivalently, in modern terms, every invertible linear transformation), is a product of elementaries.