Multilinear map with algebra

Question

I write this question "Introduction to Noncommutative Algebra" written by Bresar."6.1 Free Algebra p.139"
I didn't show that how f is multilinear.
Every F-algebra A,the map $(x_{1},...,x_{n})\longrightarrow f(x_{1},...,x_{n}),x_{i} \in A$,is multilinear.(F:filed,$1\in A$
I don't know this subject well.I know definition of multilinear polynomial,but what is multilinear map ?

Answer 1

A bilinear map is one that is linear in two inputs. In other words, \begin{align*} f(ax_1+bx_2,cx_3+dx_4)&=f(ax_1,cx_3+dx_4)+f(bx_2,cx_3+dx_4)\\ &=f(ax_1,cx_3)+f(ax_1,dx_4)+f(bx_2,cx_3)+f(bx_2,dx_4)\\ &=ac\cdot f(x_1,x_3)+ad\cdot f(x_1,x_4)+bc\cdot f(x_2,x_3)+bd\cdot f(x_2,x_4) \end{align*}

A multilinear map is one that is linear in all of its inputs (extend the properties above to arbitrary inputs). Multilinear maps are generalizations of linear maps.