Linear transformations operators in Euclidean space

Suppose that for any a we can assign, according to a certain rule, some element a. E En- We call this rule an operator A a = (a). Operator A is called linear if 

At the same time, by applying operator A to the basis vector we obtain a new vector e, which in turn can be decomposed in terms of the basis vectors  

On the other hand, we can also express vector a in the same basis by 

Comparing (A.18) with (A. 19), we find a very important relationship between the scalar components of the vectors a and a  

The matrix [Aki] is called the matrix of operator A. It describes the transformation of the scalar components of the vector by the linear operator A. 

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