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
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
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
By properties 1) and 2) we have:
This coincised with the formula for the linear transformation with standard matrix A=[[A(i,j)]]. The theorem is proved.