Here we prove the theorem about linear transformations from **R**^{n} to **R**^{m}.

Theorem. A function *f* from **R**^{n} to **R**^{m} is
a linear transformation if and only if it satisfies the following two
properties:

- For every two vectors
*A*and*B*in**R**^{n}*f(A+B)=f(A)+f(B);*

- For every vector
*A*in**R**^{n}and every number*k**f(kA)=kf(A).*

Proof.
We need to prove two statements: 1) Every linear
transformation from **R**^{n} to **R**^{m} satisfies these properties and 2) Every
function from **R**^{n} to **R**^{m} satisfying these properties is a linear transformation.

1) Suppose that *f* is a linear transformation from **R**^{n} to **R**^{m} with
standard matrix *T*. Then *f(A)=T*A* for every vector *A* in **R**^{n}. Therefore

2) Now suppose that a function *f*
from **R**^{n} to **R**^{m} satisfies properties 1 and 2. We need to prove that *f* is a
linear transformation. Consider
*n* vectors *V _{1},...,V_{n}* in

Consider now an arbitrary vector *X*=(*x _{1},...,x_{n}*). It is clear
that

By properties 1) and 2) we have:

= x

x

x

.......................

x

(A(1,1)x

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