Here we prove the theorem about linear transformations from Rn to Rm.

Theorem. A function f from Rn to Rm is a linear transformation if and only if it satisfies the following two properties:

1. For every two vectors A and B in Rn

f(A+B)=f(A)+f(B);

2. For every vector A in Rn and every number k

f(kA)=kf(A).

Proof. We need to prove two statements: 1) Every linear transformation from Rn to Rm satisfies these properties and 2) Every function from Rn to Rm satisfying these properties is a linear transformation.

1) Suppose that f is a linear transformation from Rn to Rm with standard matrix T. Then f(A)=T*A for every vector A in Rn. Therefore

f(A+B)=T(A+B)=TA+TB (by a property of matrix operations)=f(A)+f(B).
f(kA)=T*kA=kT*A (again by a property of matrix operations)=kf(A)

2) Now suppose that a function f from Rn to Rm satisfies properties 1 and 2. We need to prove that f is a linear transformation. Consider n vectors V1,...,Vn in Rn where Vi has i-th coordinate 1 and other coordinates 0. Let us defote f(Vi) by Ai=(A(1,i),...,A(m,i)).

Consider now an arbitrary vector X=(x1,...,xn). It is clear that

X=x1V1+...+xnVn

By properties 1) and 2) we have:

f(X)=f(x1V1)+...+f(xnVn)=x1f(V1)+...+xnf(Vn)
= x1A1+...xnAn =
x1(A(1,1),...,A(m,1))+
x2(A(1,2),...,A(m,2))+
.......................
xn(A(1,n),...,A(m,n))=
(A(1,1)x1+...+A(1,n)xn, A(2,1)x1+...+A(2,n)xn,...,A(m,1)x1+...+A(m,n)xn).

This coincised with the formula for the linear transformation with standard matrix A=[[A(i,j)]]. The theorem is proved.