Linear maps
A linear map is a function 
, where V and W are both vector spaces. They must satisfy the following criteria:
, for all 
, for all 
and 
Linear maps tend to preserve the properties of vector spaces under addition and scalar multiplication. A linear map is called a homomorphism of vector spaces; however, if the homomorphism is invertible (where the inverse is a homomorphism), then we call the mapping an isomorphism.
When V and W are isomorphic, we denote this as 
, and they both have the same algebraic structure.
If V and W are vector spaces in 
, and 
, then it is called a natural isomorphism. We write this as follows:
Here, 
and 
are the bases of V and W. Using the preceding equation, we can see that 
, which tells us that 
is an isomorphism.
Let's take the same vector spaces V and W as before, with bases and 
respectively. We know that is a linear map, and the matrix T that has entries Aij, where 
and 
can be defined as follows:
.
From our knowledge of matrices, we should know that the jth column of A contains Tvj in the basis of W.
Thus, 
produces a linear map 
, which we write as 
.