Linear Map, Bilinear Map, Multilinear Map

By Xah Lee. Date: 2019-03-03.

linear map 2019-02-26 kcny2 — linear map 2019-02-26

Let V and W be vector spaces over the same field K.

A function f: V → W is said to be a linear map if for any two vectors u and v both in V, and any scalar c in K, the following two conditions are satisfied:

f(u+v)=f(u)+f(v)
f(c*u)=c * f(u)

a bilinear map, is just the extension of linear map to function with more than 1 argument.

a bilinear map, is a function with 2 args, such that if one of the arg is held constant, it is a linear map.

in the same way, multilinear map is a function with n args, and when all the args are held constant except 1 of the arg, it's a linear map.