>"...these ideas were totally revolutionary, and he ended up collecting them all several years later in his groundbreaking work, Die lineale Ausdehnungslehre, e

>"...these ideas were totally revolutionary, and he ended up collecting them all several years later in his groundbreaking work, Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (https://dn790002.ca.archive.org/0/items/dielinealeausde00grasgoog/dielinealeausde00grasgoog.pdf#page=13.00 https://dn790002.ca.archive.org/0/items/dielinealeausde00gra...), published in 1844 when he was 35. Grassmann essentially came up with a totally original and new conception of linear algebra. Instead of the more traditional development of systems of linear equations, vectors, matrices, determinants, etc., Grassmann had a more general and abstract conception of the subject, which focused on what is now known as an exterior algebra, also known as the "wedge product." Unlike in the traditional development, where we use simpler operations like the dot product or cross product, in Grassmann's presentation of the subject, everything is couched in terms of the wedge product, which is a bit more abstract and harder to explain. In a nutshell, the wedge product of two vectors can be thought of as the region spanned by the vectors in space using the traditional parallelogram rule; it's basically what you probably already know as the determinant, but instead of representing the quantity of the area spanned by this space, it's the space itself. Basically, the wedge product generalizes the determinant to higher dimensions and different contexts."