# Why do we call a linear mapping "linear mapping"?

According to P. M. Cohn's Classic Algebra, for historical reasons we call a linear mapping "linear mapping". What are the historical reasons that led to the adoption of the term "linear mapping"?

The theory of Linear Algebra, along with the associated concept of linear mapping, was named as "linear" by its creator, Hermann Graßmann, which he developed in his 1844 linear algebra manifesto, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik [The Theory of Linear Extension, a New Branch of Mathematics], and also later in Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet [The Theory of Extension, Thoroughly and Rigorously Treated], which was a thorough revision of the former. With his theory of Linear Extension, Graßmann was looking to provide an algebraic framework for geometry.

In linear algebra, we deal with objects called vectors, which geometrically often represent points in space as measured with reference to some fixed point which is labelled the origin. The collection of all these vectors is called a vector space. We can perform scalar multiplication on these vectors, by stretching, shrinking, and/or reflecting them along the line passing through the origin and the point. Hence, the name of linear extension seems rather fitting. A linear map is thus a map which respects the linear structure of its domain and range, as vector spaces.

That is the historical reason we call linear maps linear. But that may not be the kind of linear map of which you may be thinking, as is evidenced by NaN's answer. Sometimes, the term linear is applied to a somewhat related, but different kind of map, namely a function of the form $ax+b$. This is linear in the linear algebra sense if and only if $b=0$. In that case all I can say is that they're probably just called linear because the graph of such a function in the plane is a straight line.

• functions of the type $ax+b$ are called affine functions. +1 For emphasising the origin of vector spaces. Jul 2 '15 at 8:05

The expression "linear mapping" can have different meaning in different contexts. In many high school textbooks $x\mapsto ax+b$ is called a "linear function". I suppose, because the graph is a straight line. According to the modern terminology, which comes from linear algebra only $x\mapsto ax$ should be called linear, while $x\mapsto ax+b$ must be called affine.

This is a technical rather than a historical reason.
In a power series like $$A + Bx + Cx^2 + \dotsb$$ you've got a constant term $A$ and a linear term $Bx$ and a quadratic term $Cx^2$. Generalizing appropriately to higher dimensions gives you the notion of constant, linear and quadratic mappings.

This seems to be the source of the term linear in linear map.

• I think "seems to be" answers need references. Jul 1 '15 at 23:49